1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,550 at ocw.mit.edu. 8 00:00:21,222 --> 00:00:23,770 GEORGE BARBASTATHIS: So I'd like to continue with what 9 00:00:23,770 --> 00:00:25,840 we were discussing last time. 10 00:00:29,620 --> 00:00:31,870 We were at the point where we're discussing the three 11 00:00:31,870 --> 00:00:35,020 dimensional wave equation. 12 00:00:35,020 --> 00:00:36,640 Before we start, let me take care 13 00:00:36,640 --> 00:00:38,680 of a couple of administrative things. 14 00:00:38,680 --> 00:00:44,050 So for one, I'm very close to finishing grading the quizzes. 15 00:00:44,050 --> 00:00:49,000 So as soon as I do, I'll put them on a DHL 16 00:00:49,000 --> 00:00:50,660 and ship them back to Boston. 17 00:00:50,660 --> 00:00:52,493 Of course, those of you who are in Singapore 18 00:00:52,493 --> 00:00:53,520 will get them from-- 19 00:00:53,520 --> 00:00:54,760 locally. 20 00:00:54,760 --> 00:00:56,770 And so that's done. 21 00:00:56,770 --> 00:00:58,720 The quiz solution, of course, has been posted. 22 00:00:58,720 --> 00:01:00,550 You must have seen it by now. 23 00:01:00,550 --> 00:01:05,332 So I would encourage you to go over this solution. 24 00:01:05,332 --> 00:01:07,540 It was a pretty good problems, actually, both of them 25 00:01:07,540 --> 00:01:09,360 were pretty good. 26 00:01:09,360 --> 00:01:11,410 And the other thing I want to say 27 00:01:11,410 --> 00:01:16,150 is that I have posted the new homework. 28 00:01:16,150 --> 00:01:18,720 So has anybody looked at it yet? 29 00:01:23,970 --> 00:01:26,870 OK, if you look at it, it might appear 30 00:01:26,870 --> 00:01:29,210 to be a little bit intimidating, especially 31 00:01:29,210 --> 00:01:30,680 the graduate version. 32 00:01:30,680 --> 00:01:35,150 It is about six or seven pages. 33 00:01:35,150 --> 00:01:38,000 But you may know from experience that if your homework is 34 00:01:38,000 --> 00:01:41,330 very wordy, it means that most of the work 35 00:01:41,330 --> 00:01:44,810 has been done for you. 36 00:01:44,810 --> 00:01:49,450 So basically, what I've done is I've-- 37 00:01:49,450 --> 00:01:52,580 the reason I-- this is the first time I give this assignment. 38 00:01:52,580 --> 00:01:55,880 And basically I develop the analogy 39 00:01:55,880 --> 00:02:01,130 between the optical Hamiltonian and the mechanical Hamiltonian. 40 00:02:01,130 --> 00:02:06,380 And I guide you through a few derivations, a few proofs 41 00:02:06,380 --> 00:02:11,358 that basically, I think, in my opinion, elucidate the analogy 42 00:02:11,358 --> 00:02:12,150 and the difference. 43 00:02:12,150 --> 00:02:14,270 Of course, they're not exactly the same. 44 00:02:14,270 --> 00:02:20,680 OK, and the reason I did it is because last year was 45 00:02:20,680 --> 00:02:23,080 the first time that they taught the Hamiltonian. 46 00:02:23,080 --> 00:02:23,840 It used to be-- 47 00:02:23,840 --> 00:02:27,590 I used to just skip that part and do other things. 48 00:02:27,590 --> 00:02:28,570 But last year I did it. 49 00:02:28,570 --> 00:02:30,028 And I got some pretty good feedback 50 00:02:30,028 --> 00:02:32,890 from the students who thought that, well, you know, this 51 00:02:32,890 --> 00:02:34,660 is kind of a cool analogy. 52 00:02:34,660 --> 00:02:36,530 So this year I expanded it a bit. 53 00:02:36,530 --> 00:02:40,670 So that's why this homework is a little bit rich. 54 00:02:40,670 --> 00:02:44,630 But it's not to bad, really, once you get into it. 55 00:02:44,630 --> 00:02:47,350 And if you follow the instructions carefully, 56 00:02:47,350 --> 00:02:50,080 each equation should taken no more than two lines. 57 00:02:50,080 --> 00:02:52,780 So it is not like a long derivation or [INAUDIBLE] 58 00:02:52,780 --> 00:02:54,230 or anything like that. 59 00:02:54,230 --> 00:02:56,623 It's just a step by step, and if you-- 60 00:02:56,623 --> 00:02:57,540 but read it carefully. 61 00:02:57,540 --> 00:03:00,678 Because there's some derivatives involved. 62 00:03:00,678 --> 00:03:03,220 So you will want to make sure you take the correct derivative 63 00:03:03,220 --> 00:03:03,850 each time. 64 00:03:03,850 --> 00:03:05,320 That's the only challenge. 65 00:03:05,320 --> 00:03:07,580 If you can manage to do it, which you will, 66 00:03:07,580 --> 00:03:10,270 if you follow the instructions carefully, 67 00:03:10,270 --> 00:03:11,710 then it should be painless. 68 00:03:11,710 --> 00:03:14,002 And there's a little bit of MATLAB that you have to do. 69 00:03:14,002 --> 00:03:15,400 But that's not valuable. 70 00:03:15,400 --> 00:03:18,640 And the code, actually the MATLAB that you have to do, 71 00:03:18,640 --> 00:03:20,580 I also give you the code. 72 00:03:20,580 --> 00:03:22,070 It's posted to the website. 73 00:03:22,070 --> 00:03:23,590 It's a bunch of functions. 74 00:03:23,590 --> 00:03:26,470 And one of the things that I do in the assignment 75 00:03:26,470 --> 00:03:31,365 is I walk you through how to use the MATLAB code. 76 00:03:31,365 --> 00:03:32,990 So what you will have to do, basically, 77 00:03:32,990 --> 00:03:35,870 is take one of the functions and modify it. 78 00:03:35,870 --> 00:03:38,023 Yeah, someone, turn on the mic. 79 00:03:38,023 --> 00:03:38,690 Have a question? 80 00:03:38,690 --> 00:03:42,750 Or-- OK. 81 00:03:46,650 --> 00:03:48,765 So going back to the-- 82 00:03:48,765 --> 00:03:49,890 I think there's a question. 83 00:03:53,170 --> 00:03:55,605 Somebody has turned on your mic, which is OK, actually, 84 00:03:55,605 --> 00:03:56,980 as long as you don't say anything 85 00:03:56,980 --> 00:03:59,080 embarrassing to [INAUDIBLE]. 86 00:04:04,170 --> 00:04:05,730 So let me go back. 87 00:04:05,730 --> 00:04:10,180 Let's go back now to the main topic. 88 00:04:10,180 --> 00:04:12,318 So, of course, we have moved on from Hamiltonians 89 00:04:12,318 --> 00:04:13,110 and all that stuff. 90 00:04:13,110 --> 00:04:15,040 We're doing wave optics now. 91 00:04:15,040 --> 00:04:19,019 And the last time we discussed the wave equation. 92 00:04:19,019 --> 00:04:20,850 And without really a proof-- 93 00:04:20,850 --> 00:04:23,910 we did the proof of the one dimensional case. 94 00:04:23,910 --> 00:04:26,580 But then without the proof, I just added. 95 00:04:26,580 --> 00:04:28,350 Basically, the one dimensional case 96 00:04:28,350 --> 00:04:33,120 was this one, just the derivative of the Z term 97 00:04:33,120 --> 00:04:35,910 without any additional terms. 98 00:04:35,910 --> 00:04:39,120 And then you can think of it simply by symmetry. 99 00:04:39,120 --> 00:04:42,010 If you accept that 3D space is symmetric, 100 00:04:42,010 --> 00:04:43,770 then the way you go from one dimension 101 00:04:43,770 --> 00:04:45,270 into the three dimensions is by just 102 00:04:45,270 --> 00:04:48,720 adding two derivatives similar to the one 103 00:04:48,720 --> 00:04:50,380 that you already had. 104 00:04:50,380 --> 00:04:52,340 And, of course, the waves that we'll get in 3D 105 00:04:52,340 --> 00:04:56,800 are much richer than the waves that we expected to get in 1D. 106 00:04:56,800 --> 00:04:59,120 And here I plotted two cases. 107 00:04:59,120 --> 00:05:02,610 One is a plane wave, which I will play the movie. 108 00:05:02,610 --> 00:05:05,070 So you will see the fringes, in this case, 109 00:05:05,070 --> 00:05:09,330 of the plane wave are moving in some arbitrary direction now. 110 00:05:09,330 --> 00:05:11,250 It can go anywhere in 3D. 111 00:05:11,250 --> 00:05:13,740 Of course, I can only show a slice 112 00:05:13,740 --> 00:05:16,050 on the plane of your projector. 113 00:05:16,050 --> 00:05:19,650 And there's a spherical wave where the wavefronts are not 114 00:05:19,650 --> 00:05:20,940 circles as you see here. 115 00:05:20,940 --> 00:05:22,470 But they're actually spheres. 116 00:05:22,470 --> 00:05:24,690 So they're spheres that are propagating 117 00:05:24,690 --> 00:05:26,940 clearly outwards in this case. 118 00:05:26,940 --> 00:05:30,450 So this is what we used to call it a diverging spherical wave 119 00:05:30,450 --> 00:05:35,130 in geometrical optics. 120 00:05:35,130 --> 00:05:37,860 Well, you can also have the opposite. 121 00:05:37,860 --> 00:05:40,590 You can have a converging spherical wave 122 00:05:40,590 --> 00:05:43,400 if the wavefronts are moving inwards. 123 00:05:43,400 --> 00:05:48,180 So one of these are valid solutions to the wave equation. 124 00:05:48,180 --> 00:05:50,470 And the plane wave is very simple. 125 00:05:50,470 --> 00:05:51,870 I will do it next. 126 00:05:51,870 --> 00:05:54,300 The spherical wave is not really difficult, 127 00:05:54,300 --> 00:05:57,350 but it involves a transformation of the coordinates 128 00:05:57,350 --> 00:06:01,322 from cartesian that we have here to polar. 129 00:06:01,322 --> 00:06:02,530 And this is done in the book. 130 00:06:02,530 --> 00:06:04,560 I'm not going to do it in the class. 131 00:06:04,560 --> 00:06:06,000 It is a little bit of work. 132 00:06:06,000 --> 00:06:09,270 And you don't get any particular intuition. 133 00:06:09,270 --> 00:06:12,090 If you take a class in electromagnetics 134 00:06:12,090 --> 00:06:16,250 or electrostatics, they will probably do it. 135 00:06:16,250 --> 00:06:18,630 But because in this class, we will 136 00:06:18,630 --> 00:06:21,360 use the spherical wave in its full form very radically-- 137 00:06:21,360 --> 00:06:23,020 we'll use an approximation to it. 138 00:06:23,020 --> 00:06:27,240 That's why I don't bother to do the full derivation. 139 00:06:27,240 --> 00:06:30,210 OK, I'm going to skip now a couple of slides. 140 00:06:30,210 --> 00:06:42,490 And I'm going to go directly to this one, slide number 16, 141 00:06:42,490 --> 00:06:46,780 which is the plane wave. 142 00:06:46,780 --> 00:06:48,820 So what I would like to do-- 143 00:06:48,820 --> 00:06:52,540 so, for the rest of the class now, pretty much for the next, 144 00:06:52,540 --> 00:06:57,770 what is it, seven or eight weeks that we have left? 145 00:06:57,770 --> 00:07:00,460 We will be talking almost exclusively 146 00:07:00,460 --> 00:07:04,720 about plane waves and their superpositions. 147 00:07:04,720 --> 00:07:07,870 Because what we will discover is that we can describe 148 00:07:07,870 --> 00:07:10,120 every optical wave and-- 149 00:07:10,120 --> 00:07:13,440 well, at least within the approximations 150 00:07:13,440 --> 00:07:15,250 that are valid in this class-- 151 00:07:15,250 --> 00:07:17,170 you can describe every optical wave 152 00:07:17,170 --> 00:07:19,290 as a superposition of plane waves. 153 00:07:19,290 --> 00:07:21,010 It's a very powerful property. 154 00:07:21,010 --> 00:07:24,720 Because plane waves, as we will see, they're very simple. 155 00:07:24,720 --> 00:07:29,210 And they obey some relatively straightforward properties. 156 00:07:29,210 --> 00:07:32,720 So if we learn how to describe those properly, 157 00:07:32,720 --> 00:07:35,620 then we can describe a great variety of other phenomena, 158 00:07:35,620 --> 00:07:39,820 including diffraction, interference, refraction, 159 00:07:39,820 --> 00:07:41,840 and a number of other things. 160 00:07:41,840 --> 00:07:44,590 So I would like to make sure that we understand plane waves. 161 00:07:44,590 --> 00:07:47,750 It is a crucial part of the class. 162 00:07:47,750 --> 00:07:52,310 OK, so in the last lecture, we wrote a simple wave 163 00:07:52,310 --> 00:07:54,650 that we'll call the harmonic wave. 164 00:07:54,650 --> 00:07:56,330 And it used to look like this. 165 00:07:56,330 --> 00:08:01,010 From now on, I will start using a phasor notation almost 166 00:08:01,010 --> 00:08:03,320 without warning. 167 00:08:03,320 --> 00:08:06,520 So when I start throwing complex exponentials, 168 00:08:06,520 --> 00:08:11,490 you know that I'm using the phasor. 169 00:08:11,490 --> 00:08:15,030 OK, so having said that, now let's 170 00:08:15,030 --> 00:08:18,480 look at the phasor of what I wrote last time. 171 00:08:18,480 --> 00:08:23,640 So last time I wrote something that looked like e to the ikz 172 00:08:23,640 --> 00:08:26,350 e to the minus i omega t. 173 00:08:26,350 --> 00:08:30,420 And I call this a harmonic wave. 174 00:08:30,420 --> 00:08:32,640 So this was in 1D. 175 00:08:32,640 --> 00:08:40,179 So the plane wave is basically the generalization of this one. 176 00:08:40,179 --> 00:08:49,010 Now, I'm going to have to write terms of the form y and x. 177 00:08:51,600 --> 00:08:54,300 Do you remember what we call the quantity k 178 00:08:54,300 --> 00:08:57,720 in the case what I still had the one dimensional wave? 179 00:08:57,720 --> 00:09:01,133 What was the name of k? 180 00:09:01,133 --> 00:09:02,050 AUDIENCE: It was the-- 181 00:09:02,050 --> 00:09:04,910 GEORGE BARBASTATHIS: Wave number, sorry, yeah, 182 00:09:04,910 --> 00:09:05,990 wave number, yeah. 183 00:09:05,990 --> 00:09:09,230 I'm sorry, someone answered in Singapore, so I-- 184 00:09:09,230 --> 00:09:10,985 OK, but he was diligent. 185 00:09:10,985 --> 00:09:12,360 So make sure to push your button. 186 00:09:15,190 --> 00:09:17,980 OK, so k is the wave number. 187 00:09:17,980 --> 00:09:23,670 And it is fair to use it if you have one dimension. 188 00:09:23,670 --> 00:09:25,330 But if you have three dimensions, 189 00:09:25,330 --> 00:09:29,340 then we actually need, at least apparently, 190 00:09:29,340 --> 00:09:32,790 we need three such quantities. 191 00:09:32,790 --> 00:09:37,010 So I need to put the z subscript here. 192 00:09:37,010 --> 00:09:41,980 And then I need two more, one for the y and one for x. 193 00:09:41,980 --> 00:09:45,965 So now the triad of these three-- 194 00:09:45,965 --> 00:09:47,590 I don't want to call them wave numbers. 195 00:09:47,590 --> 00:09:49,120 They're not really wave numbers. 196 00:09:49,120 --> 00:09:52,450 But these three quantities that have the-- 197 00:09:52,450 --> 00:09:56,500 I mean, these are essentially spatial periods, aren't they, 198 00:09:56,500 --> 00:09:58,430 in the three dimensions? 199 00:09:58,430 --> 00:10:03,070 So these quantities, if you take the triad of these quantities, 200 00:10:03,070 --> 00:10:05,050 it is called the wave vector. 201 00:10:05,050 --> 00:10:07,640 And it is usually-- 202 00:10:07,640 --> 00:10:10,270 we use the same symbol as the wave number, 203 00:10:10,270 --> 00:10:12,490 but with the vector now. 204 00:10:12,490 --> 00:10:19,270 And it consists of the three components, ky, kz. 205 00:10:19,270 --> 00:10:22,300 It is a matter of taste how you write vectors. 206 00:10:22,300 --> 00:10:23,870 In this class, I actually-- 207 00:10:23,870 --> 00:10:25,180 I will alternate a little bit. 208 00:10:25,180 --> 00:10:28,060 Sometimes I will write vectors like this 209 00:10:28,060 --> 00:10:30,100 as a triad in a parenthesis. 210 00:10:30,100 --> 00:10:32,110 Sometimes I might write the vectors 211 00:10:32,110 --> 00:10:37,870 like this where I use the unit vector, 212 00:10:37,870 --> 00:10:39,670 the unit Cartesian vectors. 213 00:10:39,670 --> 00:10:43,450 Of course-- OK, I hope that's not confusing. 214 00:10:43,450 --> 00:10:49,590 One can easily-- clearly they're both commonly used. 215 00:10:49,590 --> 00:10:52,590 So that quantity k, then, is the wave vector. 216 00:10:57,280 --> 00:11:00,305 And its significance, you can see it on the slide. 217 00:11:00,305 --> 00:11:02,680 It's probably better if I don't attempt to draw it there. 218 00:11:02,680 --> 00:11:05,380 But you can see it on the slide. 219 00:11:05,380 --> 00:11:09,580 So the wavefront of the plane wave 220 00:11:09,580 --> 00:11:16,380 is a surface which is perpendicular to the wave 221 00:11:16,380 --> 00:11:18,730 vector. 222 00:11:18,730 --> 00:11:21,100 And what is a wavefront? 223 00:11:21,100 --> 00:11:28,630 Well, wavefront, you saw it in the movie 224 00:11:28,630 --> 00:11:30,440 that I played earlier. 225 00:11:30,440 --> 00:11:36,350 Wavefront is the surface upon which the phase of the wave 226 00:11:36,350 --> 00:11:39,100 is constant. 227 00:11:39,100 --> 00:11:40,460 So what is this? 228 00:11:40,460 --> 00:11:44,450 This is basically a three dimensional sinusoid of which 229 00:11:44,450 --> 00:11:47,490 we only see a 2D slice. 230 00:11:47,490 --> 00:11:50,350 So this is sinusoid which alternates. 231 00:11:50,350 --> 00:11:54,310 The black regions are negative, minimum. 232 00:11:54,310 --> 00:11:57,560 I mean they're sort of a maximum negative value. 233 00:11:57,560 --> 00:12:02,030 The bright stripes are maximum positive value. 234 00:12:02,030 --> 00:12:08,410 And the wavefront is a surface on which the phase of the wave 235 00:12:08,410 --> 00:12:09,290 has the same value. 236 00:12:09,290 --> 00:12:11,740 For example, if the phase is zero, 237 00:12:11,740 --> 00:12:14,740 then we're talking about the positive peak. 238 00:12:14,740 --> 00:12:16,720 If their phase is pi, then we're talking 239 00:12:16,720 --> 00:12:18,700 about the negative trough, and so 240 00:12:18,700 --> 00:12:21,740 on, and so forth, or anywhere in between, for that matter. 241 00:12:21,740 --> 00:12:24,820 And the wavefront has a sense of travel to it. 242 00:12:24,820 --> 00:12:27,100 Because if I play the movie again, 243 00:12:27,100 --> 00:12:30,870 then you can see that these surfaces of constant phase 244 00:12:30,870 --> 00:12:31,810 are moving. 245 00:12:31,810 --> 00:12:36,700 This is the sense that the wave transports energy, if you wish. 246 00:12:36,700 --> 00:12:41,320 Because the wavefronts are moving with the wave speed. 247 00:12:41,320 --> 00:12:44,170 OK, so then we call it the plane wave, 248 00:12:44,170 --> 00:12:47,388 because this surface is of constant phase or planes. 249 00:12:47,388 --> 00:12:49,180 Of course, you only see them as lines here. 250 00:12:49,180 --> 00:12:52,010 Because, again, we're looking at the slice of the wave. 251 00:12:52,010 --> 00:12:53,560 But in 3D, they would be actually 252 00:12:53,560 --> 00:12:58,880 planes that would be propagating with the speed of the wave. 253 00:12:58,880 --> 00:13:01,000 So that is-- so then the wave vector, 254 00:13:01,000 --> 00:13:02,770 we need it in order to tell us what 255 00:13:02,770 --> 00:13:07,270 is the orientation of these planes. 256 00:13:07,270 --> 00:13:10,810 And for reasons of convenience, we 257 00:13:10,810 --> 00:13:15,400 define it to be perpendicular to those planes. 258 00:13:15,400 --> 00:13:20,660 Now, that's one useful thing to know. 259 00:13:20,660 --> 00:13:23,340 But that's not all. 260 00:13:23,340 --> 00:13:29,940 The wave vector, it appears to be composed of three elements. 261 00:13:29,940 --> 00:13:33,813 So the question is are these-- 262 00:13:33,813 --> 00:13:35,230 do I really have three parameters? 263 00:13:35,230 --> 00:13:38,480 Can I describe a plane wave with three parameters? 264 00:13:38,480 --> 00:13:46,890 Well, even before I do any math, you could probably guess that I 265 00:13:46,890 --> 00:13:48,300 shouldn't. 266 00:13:48,300 --> 00:13:50,730 Because these are surfaces. 267 00:13:50,730 --> 00:13:54,980 These are planar surfaces in 3D space. 268 00:13:54,980 --> 00:13:59,580 So I cannot really fit 3 dimensions of planes in a 3D 269 00:13:59,580 --> 00:14:01,087 space, right? 270 00:14:01,087 --> 00:14:02,670 Something is wrong with that argument, 271 00:14:02,670 --> 00:14:04,590 even without doing any math. 272 00:14:04,590 --> 00:14:08,740 So in actuality, these three numbers, kx, ky, and kz, 273 00:14:08,740 --> 00:14:10,680 they have to be related somehow. 274 00:14:10,680 --> 00:14:13,780 I cannot arbitrarily define all three of them. 275 00:14:13,780 --> 00:14:18,130 But I have to somehow relate them. 276 00:14:18,130 --> 00:14:19,800 So the way I relate them is I actually 277 00:14:19,800 --> 00:14:23,140 go back to my wave equation. 278 00:14:23,140 --> 00:14:26,540 And recall this thing here, the wave, 279 00:14:26,540 --> 00:14:27,940 which I will call it something. 280 00:14:27,940 --> 00:14:29,020 What did I call it? 281 00:14:29,020 --> 00:14:30,560 I didn't call it anything, actually. 282 00:14:30,560 --> 00:14:34,790 I just put an amplitude coefficient in front. 283 00:14:34,790 --> 00:14:40,210 So the amplitude coefficient, it is good for completeness, 284 00:14:40,210 --> 00:14:41,440 I guess. 285 00:14:41,440 --> 00:14:43,637 And it must satisfy the wave equation. 286 00:14:43,637 --> 00:14:44,470 So it must satisfy-- 287 00:14:47,950 --> 00:14:50,000 I'll call it something now, because-- 288 00:14:50,000 --> 00:14:51,830 f, I guess. 289 00:14:51,830 --> 00:14:57,210 So it is f of x, y, z, t. 290 00:14:57,210 --> 00:15:00,480 And it must satisfy this nasty thing, which 291 00:15:00,480 --> 00:15:09,840 looks like df dx squared plus df dy squared plus df dz squared 292 00:15:09,840 --> 00:15:18,210 minus 1 over the velocity now, d squared f over dt squared. 293 00:15:18,210 --> 00:15:19,680 It must equal to 0. 294 00:15:19,680 --> 00:15:21,650 This is what we call the wave equation. 295 00:15:21,650 --> 00:15:23,640 And for future reference, sometimes, 296 00:15:23,640 --> 00:15:28,900 instead of writing all these deduced derivatives, 297 00:15:28,900 --> 00:15:31,290 there's a shorthand which collapses 298 00:15:31,290 --> 00:15:33,575 all of this with a symbol del. 299 00:15:33,575 --> 00:15:34,950 So you can just write del squared 300 00:15:34,950 --> 00:15:39,420 f minus 1 over c squared df dt squared is 0. 301 00:15:39,420 --> 00:15:42,790 OK, that's the wave equation. 302 00:15:42,790 --> 00:15:45,560 Very good, so now what I need to do 303 00:15:45,560 --> 00:15:50,210 is I need to plug this expression into the wave 304 00:15:50,210 --> 00:15:51,038 equation. 305 00:15:51,038 --> 00:15:51,830 So what expression? 306 00:15:51,830 --> 00:15:54,050 This expression that I had for the wave, 307 00:15:54,050 --> 00:15:56,490 I need to put it into the equation. 308 00:15:56,490 --> 00:16:01,430 OK, almost by inspection, we can see what's going to happen. 309 00:16:01,430 --> 00:16:04,050 So we have an exponential, right? 310 00:16:04,050 --> 00:16:07,670 And if I take one derivative, the first derivative, 311 00:16:07,670 --> 00:16:09,680 it will knock out-- 312 00:16:09,680 --> 00:16:14,840 it will multiply by ikx, right, if I take the x derivative. 313 00:16:14,840 --> 00:16:17,000 If I take the second x derivative, 314 00:16:17,000 --> 00:16:20,120 it will bring out another ikx. 315 00:16:20,120 --> 00:16:23,660 And since i squared equals minus 1, 316 00:16:23,660 --> 00:16:28,140 it means that the derivative will look like this, d-- 317 00:16:28,140 --> 00:16:30,260 let me write it separately here. 318 00:16:30,260 --> 00:16:31,970 The second derivative, for example, 319 00:16:31,970 --> 00:16:34,460 of F with respect to x squared, it 320 00:16:34,460 --> 00:16:40,180 will equal minus kx squared times f itself, right? 321 00:16:40,180 --> 00:16:43,850 And then it can do the same for y, and z, and t. 322 00:16:43,850 --> 00:16:47,870 Except in the case of time, I will pull out omega squared. 323 00:16:47,870 --> 00:16:52,400 So if I substitute all of that now and I put it in, I will get 324 00:16:52,400 --> 00:17:00,230 that minus kx squared minus ky squared minus kz squared-- 325 00:17:00,230 --> 00:17:02,470 these are the first batch of derivatives-- 326 00:17:02,470 --> 00:17:04,790 and then minus omega squared-- 327 00:17:04,790 --> 00:17:09,560 but I have to divide by c also because c, actually, c squared 328 00:17:09,560 --> 00:17:10,910 is in the wave equation. 329 00:17:10,910 --> 00:17:15,140 So all of this, according to the wave equation, must equal to 0. 330 00:17:15,140 --> 00:17:18,200 And, of course, it is possible to satisfy this 331 00:17:18,200 --> 00:17:19,579 by setting f equal to 0. 332 00:17:19,579 --> 00:17:21,180 But that's not very interesting. 333 00:17:21,180 --> 00:17:27,560 So basically what this means, that this quantity here, it 334 00:17:27,560 --> 00:17:28,780 must equal to 0. 335 00:17:28,780 --> 00:17:36,370 So then, as advertised, the three parameters, kx, ky, kz, 336 00:17:36,370 --> 00:17:38,630 they're not independent. 337 00:17:38,630 --> 00:17:42,770 But I can specify two of them. 338 00:17:42,770 --> 00:17:48,520 The third one is completely determined by the two 339 00:17:48,520 --> 00:17:51,950 among the three plus the frequency of the wave. 340 00:17:51,950 --> 00:17:55,840 So if I give you omega, kx, and ky, 341 00:17:55,840 --> 00:18:00,920 then this equation will force kz to have a certain value. 342 00:18:00,920 --> 00:18:03,130 And you can see from this equation, which 343 00:18:03,130 --> 00:18:07,896 is written bottom of the whiteboard and on the slide-- 344 00:18:07,896 --> 00:18:09,722 AUDIENCE: [INAUDIBLE] 345 00:18:09,722 --> 00:18:11,680 GEORGE BARBASTATHIS: Oh, oh, oh, oh, thank you, 346 00:18:11,680 --> 00:18:12,950 thank you, yes, because I have a-- 347 00:18:12,950 --> 00:18:13,280 AUDIENCE: [INAUDIBLE] 348 00:18:13,280 --> 00:18:14,780 GEORGE BARBASTATHIS: Yes, thank you. 349 00:18:14,780 --> 00:18:17,180 Because I have a minus sign here. 350 00:18:17,180 --> 00:18:20,310 So the extra minus will become a plus here. 351 00:18:20,310 --> 00:18:24,220 Yes, OK, now it is correct. 352 00:18:24,220 --> 00:18:31,467 And so if you look at this equation now, what 353 00:18:31,467 --> 00:18:32,175 does it describe? 354 00:18:34,780 --> 00:18:38,620 If you think of kx, ky, kz, if you think of them now 355 00:18:38,620 --> 00:18:41,320 as a new three dimensional space, 356 00:18:41,320 --> 00:18:44,890 what is the surface that the three of them 357 00:18:44,890 --> 00:18:48,448 are constrained to lie upon? 358 00:18:48,448 --> 00:18:50,300 Button. 359 00:18:50,300 --> 00:18:51,550 AUDIENCE: A spherical shell? 360 00:18:51,550 --> 00:18:53,592 GEORGE BARBASTATHIS: It's a sphere, that's right, 361 00:18:53,592 --> 00:18:55,870 a shell, the shell of a sphere, yeah. 362 00:18:55,870 --> 00:18:58,720 And this sphere is not new to us who have seen it. 363 00:18:58,720 --> 00:19:01,660 It actually popped up in a different context. 364 00:19:01,660 --> 00:19:04,090 We called it the Descartes sphere. 365 00:19:04,090 --> 00:19:10,600 It is actually the same sphere that-- 366 00:19:10,600 --> 00:19:13,000 actually not even Descartes, an Arab scientist 367 00:19:13,000 --> 00:19:17,220 in the 11th century, guessed from Snell's law. 368 00:19:17,220 --> 00:19:21,640 So it is a little bit, I guess, magical, or soothing, 369 00:19:21,640 --> 00:19:24,940 or rewarding, whatever combination of those 370 00:19:24,940 --> 00:19:34,420 adjectives you like, that we found a nice analogy-- or not 371 00:19:34,420 --> 00:19:36,250 analogy, actually, we found an agreement 372 00:19:36,250 --> 00:19:39,400 with geometrical optics. 373 00:19:39,400 --> 00:19:44,350 In geometrical optics, we said that the radius of this sphere 374 00:19:44,350 --> 00:19:46,870 equals 2 pi upon lambda. 375 00:19:46,870 --> 00:19:49,510 This is now what, in our new language, 376 00:19:49,510 --> 00:19:53,530 we call it the wave number, times the index of refraction, 377 00:19:53,530 --> 00:19:54,340 n. 378 00:19:54,340 --> 00:19:57,770 Now this is something that I haven't told you yet. 379 00:19:57,770 --> 00:20:01,420 But I will tell you later today. 380 00:20:01,420 --> 00:20:03,940 In this equation, I haven't said anything 381 00:20:03,940 --> 00:20:05,860 about index of refraction. 382 00:20:05,860 --> 00:20:08,890 However, we know, again from geometrical optics, 383 00:20:08,890 --> 00:20:11,620 that c, the speed of light, depends 384 00:20:11,620 --> 00:20:13,130 on the index of refraction. 385 00:20:13,130 --> 00:20:16,150 So another way to derive this equation is to say-- 386 00:20:16,150 --> 00:20:21,498 to put the free space speed of light over here. 387 00:20:21,498 --> 00:20:23,540 And then in order for the equation to be correct, 388 00:20:23,540 --> 00:20:26,210 I would have to multiply by n squared, 389 00:20:26,210 --> 00:20:27,747 the index of refraction. 390 00:20:27,747 --> 00:20:29,330 So you can see from this that, indeed, 391 00:20:29,330 --> 00:20:30,757 the radius of the sphere. 392 00:20:30,757 --> 00:20:33,090 Oh, and of course, to compute the radius of this sphere, 393 00:20:33,090 --> 00:20:38,630 I have to use, once again, these dispersion relations. 394 00:20:38,630 --> 00:20:40,410 So the radius of this sphere, if you 395 00:20:40,410 --> 00:20:45,420 look from this equation over here, the radius of this sphere 396 00:20:45,420 --> 00:20:52,470 is k, the magnitude of the wave vector. 397 00:20:52,470 --> 00:21:00,420 It must equal n omega over c free space. 398 00:21:00,420 --> 00:21:02,680 OK, and now it is a matter of preference 399 00:21:02,680 --> 00:21:04,060 how you want to do it. 400 00:21:04,060 --> 00:21:08,950 For example, you can write omega as 2 pi times 401 00:21:08,950 --> 00:21:12,050 nu, where nu is of the frequency. 402 00:21:12,050 --> 00:21:15,810 Remember, omega is the angular frequency-- 403 00:21:15,810 --> 00:21:17,590 over cfs. 404 00:21:17,590 --> 00:21:19,540 Then you can use the dispersion relation, 405 00:21:19,540 --> 00:21:25,030 which says that c free space equals lambda nu. 406 00:21:25,030 --> 00:21:26,600 So nu will cancel. 407 00:21:26,600 --> 00:21:32,850 And you end up with n times 2 pi over lambda free space. 408 00:21:35,750 --> 00:21:38,737 And, of course, nu-- 409 00:21:38,737 --> 00:21:40,320 I don't say anything about free space, 410 00:21:40,320 --> 00:21:42,120 because nu is the same everywhere, 411 00:21:42,120 --> 00:21:44,400 whether it is free space, or a slave-- 412 00:21:44,400 --> 00:21:46,020 I mean, not free space. 413 00:21:46,020 --> 00:21:47,320 I'm sorry. 414 00:21:47,320 --> 00:21:49,530 OK, so that's where this equation comes from. 415 00:21:49,530 --> 00:21:52,105 That's why the radius of the k sphere, 416 00:21:52,105 --> 00:21:56,340 it equals n omega over the free-- 417 00:21:56,340 --> 00:22:00,840 over the speed of light in a vacuum or 2 pi 418 00:22:00,840 --> 00:22:03,350 n over lambda, the wavelength in free space. 419 00:22:07,040 --> 00:22:08,120 Now, those of you-- 420 00:22:08,120 --> 00:22:10,730 some of you may have taken solid state. 421 00:22:10,730 --> 00:22:14,090 This k sphere also comes up there. 422 00:22:14,090 --> 00:22:17,940 And it is called the Ewald sphere. 423 00:22:17,940 --> 00:22:21,560 In our language in optics, we seldom call it Ewald sphere. 424 00:22:21,560 --> 00:22:25,730 We call it k sphere or Descartes sphere. 425 00:22:25,730 --> 00:22:30,500 But all of these three terms, all of these three names, 426 00:22:30,500 --> 00:22:34,340 they actually denote the same thing. 427 00:22:34,340 --> 00:22:36,920 And, of course, in the case of solid state, 428 00:22:36,920 --> 00:22:44,270 this sphere does not apply to light usually. 429 00:22:44,270 --> 00:22:45,260 But it applies to what? 430 00:22:49,472 --> 00:22:52,117 Button? 431 00:22:52,117 --> 00:22:52,950 AUDIENCE: Electrons. 432 00:22:52,950 --> 00:22:55,033 GEORGE BARBASTATHIS: To electrons, that's correct. 433 00:22:55,033 --> 00:22:58,200 So according to quantum mechanics, 434 00:22:58,200 --> 00:23:00,650 electrons are also waves. 435 00:23:00,650 --> 00:23:04,230 And they satisfy-- actually that don't satisfy this equation. 436 00:23:04,230 --> 00:23:06,400 They satisfy the Schroedinger equation, 437 00:23:06,400 --> 00:23:09,690 which is in one of the problems of the homework. 438 00:23:09,690 --> 00:23:11,640 So anyway, so Schroedinger equation 439 00:23:11,640 --> 00:23:14,810 similarly leads to a sphere like this one. 440 00:23:20,830 --> 00:23:22,300 Is there one thing that-- 441 00:23:22,300 --> 00:23:26,790 OK, so this is a sort of preliminary introduction 442 00:23:26,790 --> 00:23:28,350 to plane waves. 443 00:23:28,350 --> 00:23:34,620 One thing that I would like to very strongly emphasize 444 00:23:34,620 --> 00:23:37,740 is this expression over here, the complex exponential-- 445 00:23:37,740 --> 00:23:39,690 I will rewrite it once again. 446 00:23:44,450 --> 00:23:55,410 e to the i kx x plus ky y plus kz z minus omega t, 447 00:23:55,410 --> 00:23:58,510 OK, this expression, when we see that, 448 00:23:58,510 --> 00:24:02,860 when we see an exponential with linear terms 449 00:24:02,860 --> 00:24:06,070 in the Cartesian coordinates, that's a plane wave. 450 00:24:06,070 --> 00:24:07,300 And now you know why, right? 451 00:24:07,300 --> 00:24:10,730 Because it corresponds to this moving wavefront. 452 00:24:10,730 --> 00:24:12,890 And, of course, most of the time, 453 00:24:12,890 --> 00:24:14,620 we will not add it like this. 454 00:24:14,620 --> 00:24:18,640 We will omit this term because of the phasor business 455 00:24:18,640 --> 00:24:21,070 that we discussed last time. 456 00:24:21,070 --> 00:24:23,140 So we will actually see it like this. 457 00:24:23,140 --> 00:24:27,370 And also, very often, when-- 458 00:24:27,370 --> 00:24:29,500 several things that happen in here. 459 00:24:29,500 --> 00:24:34,310 For one thing, these three numbers, as I said before, 460 00:24:34,310 --> 00:24:35,640 they're not free. 461 00:24:35,640 --> 00:24:40,250 So they are given by kx squared plus ky squared plus kz 462 00:24:40,250 --> 00:24:45,560 squared equals omega over c squared or-- 463 00:25:04,150 --> 00:25:04,660 OK. 464 00:25:04,660 --> 00:25:10,330 This is another way of solving the sphere. 465 00:25:10,330 --> 00:25:14,490 So we will see these kinds of expressions. 466 00:25:14,490 --> 00:25:17,120 For example, I might write something like e 467 00:25:17,120 --> 00:25:24,110 to the i kx x plus ky y plus. 468 00:25:34,440 --> 00:25:39,175 And there's a z here outside of the square root. 469 00:25:39,175 --> 00:25:40,550 So this looks a little bit scary. 470 00:25:40,550 --> 00:25:42,230 But it doesn't need to be scary. 471 00:25:42,230 --> 00:25:44,210 It a plane wave. 472 00:25:44,210 --> 00:25:47,550 And we will see expressions like this one later in the class. 473 00:25:47,550 --> 00:25:49,610 But I want to sort of prepare you now 474 00:25:49,610 --> 00:25:53,330 so that they are associated with a physical meaning 475 00:25:53,330 --> 00:25:54,540 in your minds. 476 00:25:54,540 --> 00:25:56,330 It is not just an expression. 477 00:25:56,330 --> 00:25:58,360 It's actually denotes a plane wave. 478 00:25:58,360 --> 00:26:01,100 The two linear terms here-- actually, all three are 479 00:26:01,100 --> 00:26:05,420 linear. x, y, and z appear as first powers. 480 00:26:05,420 --> 00:26:08,870 And the reason this square root appears here 481 00:26:08,870 --> 00:26:12,460 is because of the Descartes sphere, or the Ewald sphere, 482 00:26:12,460 --> 00:26:14,050 or whichever. 483 00:26:14,050 --> 00:26:20,838 OK, any questions about plane waves? 484 00:26:35,490 --> 00:26:39,280 The other wave that we'll be dealing with quite a bit 485 00:26:39,280 --> 00:26:41,890 is the spherical wave. 486 00:26:41,890 --> 00:26:51,420 And so clearly, as the name suggests, in this case, 487 00:26:51,420 --> 00:26:55,800 you have spherical wavefronts, which are either going away 488 00:26:55,800 --> 00:27:01,830 from a source, a diverging wave, or inwards, a converging wave. 489 00:27:01,830 --> 00:27:06,670 Just as an aside, by the way, I should 490 00:27:06,670 --> 00:27:12,910 emphasize that the spherical wave is not a physical thing. 491 00:27:12,910 --> 00:27:17,060 So plane wave is not physical either, by the way. 492 00:27:17,060 --> 00:27:18,880 Why is the plane wave not physical? 493 00:27:18,880 --> 00:27:21,880 I mean, what's an obvious reason why a plane wave must be 494 00:27:21,880 --> 00:27:23,210 an approximation of something? 495 00:27:23,210 --> 00:27:23,710 Button. 496 00:27:28,402 --> 00:27:30,360 AUDIENCE: Energy of the plane wave is infinite. 497 00:27:30,360 --> 00:27:30,660 GEORGE BARBASTATHIS: That's right. 498 00:27:30,660 --> 00:27:32,010 It has infinite energy. 499 00:27:32,010 --> 00:27:34,060 It is infinitely large. 500 00:27:34,060 --> 00:27:37,400 So, clearly, I cannot create plane waves per se. 501 00:27:37,400 --> 00:27:42,510 I can only create sort of finite approximations to plane waves. 502 00:27:42,510 --> 00:27:45,240 A spherical wave is-- 503 00:27:45,240 --> 00:27:48,300 but nevertheless, it is an interesting thing. 504 00:27:48,300 --> 00:27:50,760 And the reason we spend all this time defining it 505 00:27:50,760 --> 00:27:54,720 is because it provides a lot of mathematical convenience 506 00:27:54,720 --> 00:27:57,660 in dealing with more realistic waves, waves 507 00:27:57,660 --> 00:27:59,070 that can be realized. 508 00:27:59,070 --> 00:28:02,130 I can express them in terms of this fictitious entity, 509 00:28:02,130 --> 00:28:05,147 the plane wave. 510 00:28:05,147 --> 00:28:06,730 The same is true for the point source, 511 00:28:06,730 --> 00:28:10,150 but for a slightly different reason. 512 00:28:10,150 --> 00:28:13,690 It is actually impossible to create an ideal spherical wave. 513 00:28:13,690 --> 00:28:16,870 The best thing you can do is you can create a wave that 514 00:28:16,870 --> 00:28:20,330 would sort of, if you-- 515 00:28:22,990 --> 00:28:26,710 well, the only-- the best thing that you can do 516 00:28:26,710 --> 00:28:29,240 is you can create a dipole source here. 517 00:28:29,240 --> 00:28:31,330 So if you aim the dipole the same, 518 00:28:31,330 --> 00:28:33,760 the vertical direction, then what you can do 519 00:28:33,760 --> 00:28:37,000 is you can create a radiation pattern that 520 00:28:37,000 --> 00:28:41,590 has approximately 60, or actually, 120 degrees opening, 521 00:28:41,590 --> 00:28:43,570 126, to be-- 522 00:28:43,570 --> 00:28:45,090 63, isn't it? 523 00:28:45,090 --> 00:28:48,850 I think it's 63 degrees, the dipole. 524 00:28:48,850 --> 00:28:51,520 Anyway, so you can create a direction pattern like this, 525 00:28:51,520 --> 00:28:53,680 but never quite a spherical wave. 526 00:28:53,680 --> 00:28:56,080 In fact, someone as famous as Einstein 527 00:28:56,080 --> 00:29:00,220 said that spherical waves don't exist. 528 00:29:00,220 --> 00:29:02,170 Nevertheless, it is a very convenient 529 00:29:02,170 --> 00:29:03,430 mathematical quantity. 530 00:29:03,430 --> 00:29:06,800 And that's why we're describing it. 531 00:29:06,800 --> 00:29:11,110 But again, by the way, this is true in any kind of physics 532 00:29:11,110 --> 00:29:12,970 that you might learn. 533 00:29:12,970 --> 00:29:14,918 For example, in mechanics, very often 534 00:29:14,918 --> 00:29:16,085 you deal with frictionless-- 535 00:29:16,085 --> 00:29:16,480 AUDIENCE: [INAUDIBLE] 536 00:29:16,480 --> 00:29:17,160 GEORGE BARBASTATHIS: Is [INAUDIBLE]?? 537 00:29:17,160 --> 00:29:17,770 Yeah. 538 00:29:17,770 --> 00:29:23,330 AUDIENCE: Suppose you burst a small cracker in a water tank? 539 00:29:23,330 --> 00:29:25,720 Then it wouldn't-- wouldn't it be? 540 00:29:25,720 --> 00:29:26,910 GEORGE BARBASTATHIS: Oh, I'm sorry, I meant only in optics. 541 00:29:26,910 --> 00:29:27,200 AUDIENCE: Optics. 542 00:29:27,200 --> 00:29:28,510 GEORGE BARBASTATHIS: Yeah, in water tank, 543 00:29:28,510 --> 00:29:30,360 possibly you can create a spherical wave. 544 00:29:30,360 --> 00:29:31,818 Even so, it's not a spherical wave. 545 00:29:31,818 --> 00:29:33,010 Because it is finite, right? 546 00:29:33,010 --> 00:29:34,240 It has a finite source. 547 00:29:36,990 --> 00:29:39,240 Anyway, the reason in optics you can fundamentally not 548 00:29:39,240 --> 00:29:41,950 create this kind of thing is because of charge conservation. 549 00:29:41,950 --> 00:29:45,370 You cannot-- in order to create a spherical wave like this, 550 00:29:45,370 --> 00:29:48,300 you would have to create an oscillating charge here. 551 00:29:48,300 --> 00:29:50,470 And no such thing exists. 552 00:29:50,470 --> 00:29:53,160 So the only thing you can do is you can create a dipole. 553 00:29:53,160 --> 00:29:56,070 That is two charges which are alternating, up and down, up 554 00:29:56,070 --> 00:29:57,900 and down, positive and negative. 555 00:29:57,900 --> 00:30:02,510 That would result in a wave that it looks approximately 556 00:30:02,510 --> 00:30:05,820 like a spherical wave, but only in the direction perpendicular 557 00:30:05,820 --> 00:30:07,130 to the dipole. 558 00:30:07,130 --> 00:30:09,090 In the direction along the dipole axis, it 559 00:30:09,090 --> 00:30:10,490 actually, the wave vanishes. 560 00:30:10,490 --> 00:30:13,820 So it is very different than a spherical wave. 561 00:30:18,490 --> 00:30:22,360 To justify it a little bit why, even though it is non-physical, 562 00:30:22,360 --> 00:30:25,550 why we spend so much time discussing it, 563 00:30:25,550 --> 00:30:30,060 it has to do, actually, with systems theory. 564 00:30:30,060 --> 00:30:31,780 Yes, question? 565 00:30:31,780 --> 00:30:34,180 AUDIENCE: Isn't a dipole actually canceling out 566 00:30:34,180 --> 00:30:40,110 because of the distance between the top and bottom source? 567 00:30:40,110 --> 00:30:42,300 You have like a different distance between them. 568 00:30:42,300 --> 00:30:44,430 GEORGE BARBASTATHIS: The charge, the charge is 0. 569 00:30:44,430 --> 00:30:45,097 The charge is 0. 570 00:30:45,097 --> 00:30:46,555 But you still have a dipole moment. 571 00:30:46,555 --> 00:30:48,840 Then moreover, if the dipole is oscillating, that is, 572 00:30:48,840 --> 00:30:50,700 if it is going from positive to negative, 573 00:30:50,700 --> 00:30:53,700 positive to negative very, very fast, then you 574 00:30:53,700 --> 00:30:54,870 have charges accelerating. 575 00:30:54,870 --> 00:30:57,440 So therefore you generate electromagnetic radiation, 576 00:30:57,440 --> 00:30:58,790 as we will see in a little bit. 577 00:30:58,790 --> 00:31:01,920 AUDIENCE: No, no, I mean the intensity 578 00:31:01,920 --> 00:31:05,580 of the field from the dipole isn't necessarily 579 00:31:05,580 --> 00:31:09,240 vanishing along the axis if the spacing's set the right way. 580 00:31:12,945 --> 00:31:15,070 GEORGE BARBASTATHIS: Not everywhere along the axis, 581 00:31:15,070 --> 00:31:17,112 but you will definitely get nulls along the axis. 582 00:31:22,870 --> 00:31:24,755 If I remember correctly, yeah, the dipole 583 00:31:24,755 --> 00:31:26,630 has a radiation pattern that looks like this. 584 00:31:26,630 --> 00:31:28,370 It looks like a butterfly. 585 00:31:28,370 --> 00:31:31,200 AUDIENCE: Yeah, but it's dependent on the distance 586 00:31:31,200 --> 00:31:34,500 between the plus and the minus charges. 587 00:31:34,500 --> 00:31:36,000 GEORGE BARBASTATHIS: No, I'm talking 588 00:31:36,000 --> 00:31:38,340 about an infinitesimally small dipole now. 589 00:31:38,340 --> 00:31:39,132 AUDIENCE: OK, cool. 590 00:31:39,132 --> 00:31:40,507 GEORGE BARBASTATHIS: Yeah, if you 591 00:31:40,507 --> 00:31:42,255 have like a lambda over four and 10 592 00:31:42,255 --> 00:31:43,660 or something like that, then yes, 593 00:31:43,660 --> 00:31:46,170 the radiation pattern looks different. 594 00:31:46,170 --> 00:31:49,500 But in the simplest possible case that you can approximate 595 00:31:49,500 --> 00:31:53,730 the best spherical wave is an infinitesimally small dipole 596 00:31:53,730 --> 00:31:56,950 whose radiation pattern looks like a butterfly. 597 00:32:02,727 --> 00:32:05,060 So we actually went a little bit ahead of ourselves now. 598 00:32:05,060 --> 00:32:06,840 Because this entire discussion, it 599 00:32:06,840 --> 00:32:09,350 involved electromagnetics, which we haven't done yet. 600 00:32:09,350 --> 00:32:12,360 But by the end of this lecture, it 601 00:32:12,360 --> 00:32:13,860 will be a little bit more clear what 602 00:32:13,860 --> 00:32:16,970 I mean by dipoles and all of these things. 603 00:32:16,970 --> 00:32:20,870 The thing I meant to say is that the spherical wave, 604 00:32:20,870 --> 00:32:25,300 mathematically, it corresponds to a point source. 605 00:32:25,300 --> 00:32:30,230 And that is something that most of you, probably, 606 00:32:30,230 --> 00:32:34,430 at some point or another, took a class on linear systems 607 00:32:34,430 --> 00:32:40,725 where typically you learn a term called an impulse response. 608 00:32:40,725 --> 00:32:42,350 So the impulse pulse in the time domain 609 00:32:42,350 --> 00:32:45,850 is a very narrow excitation. 610 00:32:45,850 --> 00:32:48,700 In fact, it is infinitesimally narrow. 611 00:32:48,700 --> 00:32:50,360 And it is so narrow that, because it 612 00:32:50,360 --> 00:32:53,570 has to carry finite energy, it also has infinite amplitude. 613 00:32:53,570 --> 00:32:55,680 That's what it's called an impulse. 614 00:32:55,680 --> 00:32:58,290 So the spherical wave is the equivalent of that, 615 00:32:58,290 --> 00:32:59,840 but in space now. 616 00:32:59,840 --> 00:33:03,920 The spherical wave originates as an infinitesimally small 617 00:33:03,920 --> 00:33:09,200 disturbance, if you wish, which then mathematically creates 618 00:33:09,200 --> 00:33:15,552 this uniformly or isotropically expanding spherical wavefront. 619 00:33:15,552 --> 00:33:17,510 And from that point of view, it is very useful. 620 00:33:17,510 --> 00:33:20,210 Because even the dipole that I said before, 621 00:33:20,210 --> 00:33:22,700 it can be described, actually, as a superposition 622 00:33:22,700 --> 00:33:24,730 of a spherical wave. 623 00:33:24,730 --> 00:33:34,320 So I can be physically correct if I do my physics correct. 624 00:33:34,320 --> 00:33:37,000 I can get the proper answer. 625 00:33:37,000 --> 00:33:39,990 So the spherical wave provides this mathematical convenience. 626 00:33:39,990 --> 00:33:42,530 Also, to be honest, the math that we will do here, 627 00:33:42,530 --> 00:33:44,310 it is a very good approximation to what 628 00:33:44,310 --> 00:33:45,750 you observe in the laboratory. 629 00:33:45,750 --> 00:33:48,750 That's another good justification for using it. 630 00:33:48,750 --> 00:33:50,760 Because even though we know that there 631 00:33:50,760 --> 00:33:53,790 is no such thing as a point source, 632 00:33:53,790 --> 00:33:56,670 many of the things that we will derive in the next two 633 00:33:56,670 --> 00:34:01,030 lectures based in this sort of crude approximation, 634 00:34:01,030 --> 00:34:03,900 they tend now to be very good approximations for what 635 00:34:03,900 --> 00:34:05,400 you see in the laboratory. 636 00:34:05,400 --> 00:34:08,280 For example, if you take a pinhole, 637 00:34:08,280 --> 00:34:09,900 a pinhole is a finite thing, right? 638 00:34:09,900 --> 00:34:12,480 I mean, the smallest pinhole that people 639 00:34:12,480 --> 00:34:16,139 use in the laboratory, it may have a diameter of one micron. 640 00:34:16,139 --> 00:34:18,449 Sometimes we use bigger, three, five micron, 641 00:34:18,449 --> 00:34:19,429 something like that. 642 00:34:19,429 --> 00:34:21,179 Well, if you pass light through a pinhole, 643 00:34:21,179 --> 00:34:23,610 the light that comes out is, to a very good approximation, 644 00:34:23,610 --> 00:34:27,070 described by a spherical wave, but of course not everywhere, 645 00:34:27,070 --> 00:34:27,570 right? 646 00:34:27,570 --> 00:34:30,250 The light coming out of a pinhole, 647 00:34:30,250 --> 00:34:31,570 it doesn't come out backwards. 648 00:34:31,570 --> 00:34:33,250 It only goes forward, right? 649 00:34:33,250 --> 00:34:37,409 So therefore, you have-- you know, clearly, the expression 650 00:34:37,409 --> 00:34:39,600 is approximately correct, but only 651 00:34:39,600 --> 00:34:44,300 for the part of the wave, the portion that is physical. 652 00:34:44,300 --> 00:34:47,070 So we have to be a little bit mindful of what 653 00:34:47,070 --> 00:34:50,310 is the meaning of everything that we write down, 654 00:34:50,310 --> 00:34:55,014 especially when things like that are involved. 655 00:34:55,014 --> 00:34:56,639 The spherical wave has another problem. 656 00:34:56,639 --> 00:35:01,150 It has the problem that it blows up at the center. 657 00:35:01,150 --> 00:35:04,140 And the reason it blows up is because-- 658 00:35:04,140 --> 00:35:07,350 well, like the impulse in the time domain 659 00:35:07,350 --> 00:35:12,980 that you learned in linear dynamical systems, 660 00:35:12,980 --> 00:35:15,720 it has to have a finite energy. 661 00:35:15,720 --> 00:35:18,000 So in this context here, the finite energy, 662 00:35:18,000 --> 00:35:21,930 as you proved in the homework some time ago, 663 00:35:21,930 --> 00:35:25,260 it means that the amplitude of this spherical wave 664 00:35:25,260 --> 00:35:29,100 must decrease like one over the distance. 665 00:35:29,100 --> 00:35:32,050 So this 1 over r that we get in the denominator, of course, 666 00:35:32,050 --> 00:35:33,950 at r equals 0, it explodes, right? 667 00:35:33,950 --> 00:35:39,020 So, again, that is the delta function, the impulse analogy. 668 00:35:39,020 --> 00:35:42,650 So the spherical wave has a few mathematical inconveniences 669 00:35:42,650 --> 00:35:44,810 of this sort that we have to be mindful with. 670 00:35:49,570 --> 00:35:53,900 Now, there's one more mysterious thing about it, 671 00:35:53,900 --> 00:35:55,270 which I will just point out. 672 00:35:55,270 --> 00:35:59,372 And I will then move on without saying why. 673 00:35:59,372 --> 00:36:01,580 If you look at the expression of the spherical wave-- 674 00:36:01,580 --> 00:36:02,455 let me write it down. 675 00:36:02,455 --> 00:36:16,770 It looks like cosine kr minus omega t over r. 676 00:36:16,770 --> 00:36:21,040 And let's take stock here. 677 00:36:21,040 --> 00:36:22,570 So the 1 over r we discussed. 678 00:36:22,570 --> 00:36:24,330 The 1 over r is energy conservation. 679 00:36:24,330 --> 00:36:26,280 We always like that. 680 00:36:26,280 --> 00:36:30,030 The cosine that you see in the numerator, 681 00:36:30,030 --> 00:36:39,514 if I go back and play the movie of the spherical wave, 682 00:36:39,514 --> 00:36:42,910 we have to wait for the plane wave to finish here. 683 00:36:42,910 --> 00:36:45,900 So the kr minus omega t, it is this sort 684 00:36:45,900 --> 00:36:50,770 of explosion of the wavefronts that you see in the movie. 685 00:36:50,770 --> 00:36:57,110 So kr, if i is the radius, kr describes this sphere again. 686 00:36:57,110 --> 00:37:00,520 So the cosine kr basically tells you 687 00:37:00,520 --> 00:37:03,880 that you have these alternating positive and negative 688 00:37:03,880 --> 00:37:06,100 valued spheres. 689 00:37:06,100 --> 00:37:08,500 And the minus omega t is the traveling aspect. 690 00:37:08,500 --> 00:37:11,140 It tells you that these spheres explode. 691 00:37:11,140 --> 00:37:14,960 They propagate outwards as a function of time. 692 00:37:14,960 --> 00:37:18,800 So this is the meaning of the kr minus omega t. 693 00:37:18,800 --> 00:37:20,980 Then it bothers to put another term over there, 694 00:37:20,980 --> 00:37:23,350 minus pi over 2. 695 00:37:23,350 --> 00:37:25,450 And this might sound a little bit weird. 696 00:37:25,450 --> 00:37:30,460 I mean, in general, you can put any phase that you like there. 697 00:37:30,460 --> 00:37:36,760 It means that you have a shifted time axis. 698 00:37:36,760 --> 00:37:38,200 So I could put 5 there. 699 00:37:38,200 --> 00:37:39,790 Why did they bother to put pi over 2? 700 00:37:39,790 --> 00:37:42,460 What is the pi over 2 relative to? 701 00:37:42,460 --> 00:37:47,240 OK, it turns out if you properly solve Maxwell's equations, 702 00:37:47,240 --> 00:37:49,000 which we haven't seen yet, but I'm 703 00:37:49,000 --> 00:37:51,760 sure you are aware that there is such a thing called 704 00:37:51,760 --> 00:37:54,610 Maxwell's equation that describes 705 00:37:54,610 --> 00:37:55,680 electromagnetic waves. 706 00:37:55,680 --> 00:38:01,210 OK, so if you solve it properly with this kind of point source, 707 00:38:01,210 --> 00:38:05,360 because of something having to do with magnetic fields, 708 00:38:05,360 --> 00:38:07,990 you pick up a phase delay with respect to what? 709 00:38:07,990 --> 00:38:10,730 You pick up a phase delay with respect to the source. 710 00:38:10,730 --> 00:38:12,070 So this is now really weird. 711 00:38:12,070 --> 00:38:14,980 It tells you if you have a source that is oscillating, 712 00:38:14,980 --> 00:38:19,450 let's say, like a cosine, so it is maximum at t equals 0, 713 00:38:19,450 --> 00:38:22,240 the wave that comes out will actually be a sine. 714 00:38:22,240 --> 00:38:24,910 So it will be phase shifted by pi over 2. 715 00:38:24,910 --> 00:38:29,670 There's no easy way to explain this either than solving 716 00:38:29,670 --> 00:38:30,910 Maxwell's equation. 717 00:38:30,910 --> 00:38:35,120 There is actually some papers in the literature-- this is very, 718 00:38:35,120 --> 00:38:37,230 very interesting for an-- 719 00:38:37,230 --> 00:38:38,520 I don't know. 720 00:38:38,520 --> 00:38:40,277 it's strange, right? 721 00:38:40,277 --> 00:38:41,860 It can be explained from the equation, 722 00:38:41,860 --> 00:38:43,460 but it's still a little bit strange. 723 00:38:43,460 --> 00:38:46,380 So people are actually working on this phase shift. 724 00:38:46,380 --> 00:38:49,390 There's some very strange aspects to it. 725 00:38:49,390 --> 00:38:52,450 Anyway, for us, the phase shift, we'll just accept it. 726 00:38:52,450 --> 00:38:53,860 It is well established. 727 00:38:53,860 --> 00:38:57,850 So we'll just accept it and move on. 728 00:38:57,850 --> 00:39:00,690 When we write it as a phasor, I mean 729 00:39:00,690 --> 00:39:03,700 in the complex presentation, of course, 730 00:39:03,700 --> 00:39:08,180 e to the minus i pi over 2 is minus i. 731 00:39:08,180 --> 00:39:10,760 So I skipped a few steps of the slide. 732 00:39:10,760 --> 00:39:12,690 So let me do them on the whiteboard. 733 00:39:12,690 --> 00:39:15,320 So to go from this in the complex presentation, 734 00:39:15,320 --> 00:39:21,350 you would write e to the i, kr minus omega t minus pi over 2. 735 00:39:21,350 --> 00:39:23,930 And, of course, you still have this thing outside. 736 00:39:23,930 --> 00:39:28,300 Nothing can-- this is not affected. 737 00:39:28,300 --> 00:39:30,890 OK, then remember that e to the minus 738 00:39:30,890 --> 00:39:34,820 i pi over 2 equals minus i. 739 00:39:34,820 --> 00:39:36,590 And it equals 1 over i. 740 00:39:36,590 --> 00:39:41,380 So this is why the i appeared in the denominator over here. 741 00:39:41,380 --> 00:39:44,010 So because pi over 2 is kind of a pain 742 00:39:44,010 --> 00:39:46,520 to write down, when we write the spherical way, 743 00:39:46,520 --> 00:39:50,130 we'll just write it with the i in the denominator. 744 00:39:50,130 --> 00:39:51,600 And this i is actually not-- 745 00:39:51,600 --> 00:39:54,700 I mean, it's not a big deal, you know. 746 00:39:54,700 --> 00:40:00,910 If you-- in fact, myself, sometimes I will just skip it. 747 00:40:00,910 --> 00:40:03,640 But it's kind of nice to know that it's there. 748 00:40:06,240 --> 00:40:08,830 And, of course, in the-- the other thing 749 00:40:08,830 --> 00:40:10,590 that we'll do commonly, but by now 750 00:40:10,590 --> 00:40:13,260 a thing where we're comfortable with that 751 00:40:13,260 --> 00:40:19,110 is that we neglect the temporal variation. 752 00:40:19,110 --> 00:40:20,360 And then we'll get the phasor. 753 00:40:24,000 --> 00:40:26,880 The last thing that we'll do before moving on to a slightly 754 00:40:26,880 --> 00:40:30,150 different topic is-- 755 00:40:30,150 --> 00:40:32,010 remember in geometrical optics, we 756 00:40:32,010 --> 00:40:34,650 derived a lot of useful things using 757 00:40:34,650 --> 00:40:36,750 this paraxial approximation. 758 00:40:36,750 --> 00:40:39,700 So the paraxial approximation basically says that-- 759 00:40:43,020 --> 00:40:48,330 in this case, the wave originated relatively far away. 760 00:40:48,330 --> 00:40:50,370 And you have these spherical wavefronts 761 00:40:50,370 --> 00:40:53,850 which are nice spheres near the source. 762 00:40:53,850 --> 00:40:59,270 But if you go at a relatively long distance, basically, 763 00:40:59,270 --> 00:41:01,730 you cannot tell the difference between these spherical 764 00:41:01,730 --> 00:41:04,490 wavefronts and parabolas. 765 00:41:04,490 --> 00:41:06,830 And it turns out that, because the sphere involves 766 00:41:06,830 --> 00:41:09,860 this nasty square root, it is much more 767 00:41:09,860 --> 00:41:12,820 convenient to represent this as parabolas. 768 00:41:12,820 --> 00:41:15,430 So the way you go from the sphere to the parabola, 769 00:41:15,430 --> 00:41:16,402 I did it on the slide. 770 00:41:16,402 --> 00:41:17,360 And I will do it again. 771 00:41:17,360 --> 00:41:19,610 Because this is a derivation I want 772 00:41:19,610 --> 00:41:21,860 you to be very familiar with. 773 00:41:21,860 --> 00:41:23,060 We've done it a few times. 774 00:41:23,060 --> 00:41:25,220 And we'll do it again and again. 775 00:41:25,220 --> 00:41:28,920 You start with the expression for the wave. 776 00:41:28,920 --> 00:41:32,000 So it is e to the ikr, right? 777 00:41:32,000 --> 00:41:33,620 And there is bells and whistles. 778 00:41:33,620 --> 00:41:34,533 There is all the r's. 779 00:41:34,533 --> 00:41:35,200 There's the i's. 780 00:41:35,200 --> 00:41:36,990 Let me just write this. 781 00:41:36,990 --> 00:41:40,550 OK, and then remember, so we have r. 782 00:41:40,550 --> 00:41:42,620 This is the polar coordinate. 783 00:41:42,620 --> 00:41:47,280 So it is x plus y plus z squared. 784 00:41:47,280 --> 00:41:49,370 And the paraxial approximation-- 785 00:41:49,370 --> 00:41:51,800 let me put this back-- 786 00:41:51,800 --> 00:41:57,300 in our convention, usually, is that this is the z-axis. 787 00:41:59,870 --> 00:42:04,110 So z is the axis of propagation, the optical axis. 788 00:42:04,110 --> 00:42:07,060 And then you have the other two axes perpendicular to it. 789 00:42:07,060 --> 00:42:10,260 Typically, we'll denote x as being vertical. 790 00:42:10,260 --> 00:42:13,880 OK, and is this now right handed, the way I did it? 791 00:42:13,880 --> 00:42:16,080 Yeah, right hand, OK good. 792 00:42:16,080 --> 00:42:18,650 All right, so the paraxial approximation 793 00:42:18,650 --> 00:42:22,910 then means that z is much bigger than the values 794 00:42:22,910 --> 00:42:24,320 of the other two coordinates. 795 00:42:24,320 --> 00:42:26,660 Because, basically, I'm limiting myself 796 00:42:26,660 --> 00:42:29,120 to operate in this region. 797 00:42:29,120 --> 00:42:31,800 that's the meaning of the paraxial approximation. 798 00:42:31,800 --> 00:42:34,100 So what you do then is you do a Taylor expansion 799 00:42:34,100 --> 00:42:35,840 on this square root. 800 00:42:35,840 --> 00:42:37,880 So you pull z outside. 801 00:42:46,290 --> 00:42:51,660 Then you use the Taylor property that says 1 plus small 802 00:42:51,660 --> 00:42:55,380 is approximately equal to 1 plus small over 2. 803 00:42:55,380 --> 00:42:59,320 And then you can see that r equals 804 00:42:59,320 --> 00:43:07,750 z 1 plus x squared plus y squared over z squared times 2, 805 00:43:07,750 --> 00:43:09,945 which is also known as z plus-- 806 00:43:15,950 --> 00:43:16,540 OK. 807 00:43:16,540 --> 00:43:18,370 So that's the paraxial approximation 808 00:43:18,370 --> 00:43:21,750 for the polar variable. 809 00:43:21,750 --> 00:43:26,020 And now I can, of course, put it in my expression. 810 00:43:26,020 --> 00:43:30,750 So this means that e to the ikr is approximately equal 811 00:43:30,750 --> 00:43:40,590 to e to the ikz, e to the ik, x squared plus y squared over 2z. 812 00:43:40,590 --> 00:43:43,650 And sometimes we will leave it like this. 813 00:43:43,650 --> 00:43:51,780 Or sometimes, we will remember that k equals 2 pi upon lambda. 814 00:43:51,780 --> 00:43:58,830 And then this expression now becomes e to the i 2 pi 815 00:43:58,830 --> 00:44:00,900 z upon lambda. 816 00:44:00,900 --> 00:44:02,730 e to the i, notice what will happen. 817 00:44:02,730 --> 00:44:05,640 The 2s will cancel. 818 00:44:05,640 --> 00:44:07,290 A pi will remain. 819 00:44:07,290 --> 00:44:08,370 And you'll get-- 820 00:44:12,600 --> 00:44:13,960 OK. 821 00:44:13,960 --> 00:44:16,520 So all of these are interchangeable expressions 822 00:44:16,520 --> 00:44:19,590 that we can use for spherical waves. 823 00:44:23,720 --> 00:44:26,580 Another interesting question-- I omitted something. 824 00:44:26,580 --> 00:44:30,262 I omitted 1 over ir. 825 00:44:30,262 --> 00:44:31,970 And actually, there's also the amplitude. 826 00:44:31,970 --> 00:44:35,230 Let's put the amplitude for completeness. 827 00:44:35,230 --> 00:44:37,560 What happens to the ir? 828 00:44:37,560 --> 00:44:42,240 Should they also do some paraxial approximation over it? 829 00:44:42,240 --> 00:44:54,840 It turns out it is sufficient to simply approximate it as z. 830 00:44:54,840 --> 00:45:00,000 For the part that appears outside of the exponent, 831 00:45:00,000 --> 00:45:02,850 we can simply approximate it as z. 832 00:45:02,850 --> 00:45:06,390 And the reason is because r, it is actually slowly varying. 833 00:45:06,390 --> 00:45:08,720 You know, if you compare-- 834 00:45:08,720 --> 00:45:11,920 what is r? 835 00:45:11,920 --> 00:45:13,150 Let's see if we can use the-- 836 00:45:13,150 --> 00:45:17,733 yeah, so r is this, right? 837 00:45:17,733 --> 00:45:18,525 So if you compare-- 838 00:45:35,450 --> 00:45:39,910 OK, so basically, you have to pick z equals constant. 839 00:45:39,910 --> 00:45:42,240 So I have to pick a plane that is 840 00:45:42,240 --> 00:45:44,760 tangential to the sphere at this place. 841 00:45:44,760 --> 00:45:48,990 So I compare r1 to r2, right? 842 00:45:48,990 --> 00:45:53,710 So r1 goes all the way to the plane. r2 simply stops here. 843 00:45:53,710 --> 00:45:56,530 So think about it this way. 844 00:45:56,530 --> 00:46:01,780 Let's say that the error that they make-- 845 00:46:01,780 --> 00:46:05,460 when I say z is approximately equal to r, 846 00:46:05,460 --> 00:46:07,163 let's say that they make a small error. 847 00:46:07,163 --> 00:46:08,830 Let's say that they make an error of 1%. 848 00:46:12,360 --> 00:46:16,770 That means that they misrepresent the amplitude 849 00:46:16,770 --> 00:46:19,850 of the wave by 1%. 850 00:46:19,850 --> 00:46:25,680 Now, let's look at the term that is inside the exponential, 851 00:46:25,680 --> 00:46:30,080 e to the i pi xy, x squared plus y squared over lambda z. 852 00:46:33,470 --> 00:46:41,320 If I miss by 1%, that is, if I had e to the 1,000 pi 853 00:46:41,320 --> 00:46:50,270 and I miss by 1%, then it means I went to e to the 990 pi, 854 00:46:50,270 --> 00:46:51,100 right? 855 00:46:51,100 --> 00:46:55,060 So this 1% error actually caused me, 856 00:46:55,060 --> 00:46:59,410 how many, 10 full oscillations. 857 00:46:59,410 --> 00:47:02,680 That's why you have to keep a better approximation 858 00:47:02,680 --> 00:47:09,840 inside the exponential than I can afford to keep outside. 859 00:47:09,840 --> 00:47:15,360 So with the phase, I have to be much more accurate in order 860 00:47:15,360 --> 00:47:18,120 to make sure that I don't make a big error in the phase delay. 861 00:47:18,120 --> 00:47:21,510 And that's because the cosine varies very rapidly. 862 00:47:21,510 --> 00:47:25,170 As you increase r, the cosine is oscillating. 863 00:47:25,170 --> 00:47:28,710 But 1 over r is a slowly, what we call a slowly varying 864 00:47:28,710 --> 00:47:30,270 function of r. 865 00:47:30,270 --> 00:47:33,540 And therefore, we can afford to be sloppier with it, 866 00:47:33,540 --> 00:47:34,620 if you wish. 867 00:47:34,620 --> 00:47:37,770 So the paraxial approximation, then, for the spherical wave 868 00:47:37,770 --> 00:47:45,850 is just like that with z in the denominator 869 00:47:45,850 --> 00:47:50,523 but this additional quadratic term in the exponent. 870 00:47:50,523 --> 00:47:52,440 And, of course, this additional quadratic term 871 00:47:52,440 --> 00:47:55,770 is the one where all of the action is happening. 872 00:47:55,770 --> 00:47:58,890 And this-- we will see this later in the class. 873 00:47:58,890 --> 00:48:00,960 And that's the second important lesson 874 00:48:00,960 --> 00:48:07,160 from today, which is that if we see an expression which 875 00:48:07,160 --> 00:48:12,498 is quadratic in the Cartesian coordinates, 876 00:48:12,498 --> 00:48:13,790 we'll call it a spherical wave. 877 00:48:16,550 --> 00:48:20,570 So we went through all these adventures, I guess, 878 00:48:20,570 --> 00:48:25,210 to arrive at two very basic waves. 879 00:48:25,210 --> 00:48:31,060 One is the kind of wave in whose phasor you have linear terms 880 00:48:31,060 --> 00:48:33,520 in the cartesian coordinates. 881 00:48:33,520 --> 00:48:34,840 That we'll call the plane wave. 882 00:48:47,970 --> 00:48:51,895 OK, so if you see something like this, 883 00:48:51,895 --> 00:48:54,020 it doesn't really matter what these things are, kx, 884 00:48:54,020 --> 00:48:55,650 ky, and so on. 885 00:48:55,650 --> 00:49:01,110 But because these terms are linear, it is plane. 886 00:49:01,110 --> 00:49:04,030 When you arrive at an expression that looks like this, 887 00:49:04,030 --> 00:49:07,630 you might have all kinds of junk. 888 00:49:07,630 --> 00:49:11,770 And somewhere you have something of the form e to the i pi 889 00:49:11,770 --> 00:49:18,150 x squared plus y squared over lambda z, 890 00:49:18,150 --> 00:49:29,100 this quadratic means that it is spherical, or, more accurately, 891 00:49:29,100 --> 00:49:32,280 the paraxial approximation to a spherical wave. 892 00:49:37,620 --> 00:49:40,290 Yes, button. 893 00:49:40,290 --> 00:49:44,150 AUDIENCE: I'm curious what would be the propagation 894 00:49:44,150 --> 00:49:46,155 of an evanescent wave? 895 00:49:46,155 --> 00:49:48,530 GEORGE BARBASTATHIS: Let's postpone that for a little bit 896 00:49:48,530 --> 00:49:50,392 later. 897 00:49:50,392 --> 00:49:51,350 That's a good question. 898 00:49:51,350 --> 00:49:55,260 But we don't have the tools to deal with that yet. 899 00:49:55,260 --> 00:49:56,630 [INAUDIBLE] is asking-- 900 00:49:56,630 --> 00:49:57,880 I wrote this square root here. 901 00:50:02,090 --> 00:50:02,590 Here it is. 902 00:50:05,970 --> 00:50:10,120 [INAUDIBLE] is asking, what if I pick kx and ky 903 00:50:10,120 --> 00:50:12,830 so that the argument of the square root becomes negative? 904 00:50:12,830 --> 00:50:16,220 Therefore, this quantity would become imaginary, right? 905 00:50:16,220 --> 00:50:18,710 That is called an evanescent wave. 906 00:50:18,710 --> 00:50:21,330 But let's not go into it now. 907 00:50:21,330 --> 00:50:24,113 Yeah, we'll come back to it though. 908 00:50:24,113 --> 00:50:25,030 It's a valid question. 909 00:50:32,000 --> 00:50:33,110 And other questions? 910 00:50:38,783 --> 00:50:41,200 Or I should say, any questions that I'm willing to answer? 911 00:50:41,200 --> 00:50:42,825 Because that is a good question, but--. 912 00:50:49,920 --> 00:50:56,180 OK, did you come up with any questions about spherical waves 913 00:50:56,180 --> 00:51:00,340 or plane waves during the break? 914 00:51:00,340 --> 00:51:02,200 AUDIENCE: OK, here I have one question. 915 00:51:02,200 --> 00:51:05,530 So just now we mentioned that when the kx, ky is 916 00:51:05,530 --> 00:51:10,510 very small while the kz is very big, we call it plane wave, 917 00:51:10,510 --> 00:51:11,050 correct? 918 00:51:11,050 --> 00:51:14,832 GEORGE BARBASTATHIS: Not quite. 919 00:51:14,832 --> 00:51:16,290 It is true, actually, what you say. 920 00:51:16,290 --> 00:51:18,020 kx and ky will be very small. 921 00:51:21,097 --> 00:51:21,930 AUDIENCE: Yeah, at-- 922 00:51:21,930 --> 00:51:23,220 AUDIENCE: [INAUDIBLE] 923 00:51:23,220 --> 00:51:24,350 GEORGE BARBASTATHIS: Yeah. 924 00:51:24,350 --> 00:51:25,933 AUDIENCE: OK, but this one I'm talking 925 00:51:25,933 --> 00:51:27,410 about in the k dimension. 926 00:51:27,410 --> 00:51:30,390 How about in the space, in the real 3D space. 927 00:51:30,390 --> 00:51:32,640 GEORGE BARBASTATHIS: OK, the paraxial approximation, I 928 00:51:32,640 --> 00:51:34,500 did actually in space. 929 00:51:34,500 --> 00:51:41,010 So this diagram over here, that was space, x, y, z. 930 00:51:41,010 --> 00:51:44,730 But what you say is true in the spherical wave. 931 00:51:47,360 --> 00:51:50,027 The reason I'm reluctant to admit what you say 932 00:51:50,027 --> 00:51:52,110 is because the spherical wave is not a plane wave. 933 00:51:52,110 --> 00:51:56,950 So therefore kx, ky, kz, they vary across the wavefront, 934 00:51:56,950 --> 00:51:57,637 right? 935 00:51:57,637 --> 00:51:59,470 So it's a bit more complicated what happens. 936 00:51:59,470 --> 00:52:04,350 But yeah, if you took a small portion of the wavefront here, 937 00:52:04,350 --> 00:52:09,420 and you plotted kx-- 938 00:52:09,420 --> 00:52:12,060 well, this is the direction. 939 00:52:12,060 --> 00:52:14,070 If you plotted, kz would be big. 940 00:52:14,070 --> 00:52:16,580 And then kx would be very small. 941 00:52:16,580 --> 00:52:18,958 So it is true. kx and ky are also very small. 942 00:52:18,958 --> 00:52:21,000 But you have to be careful with a spherical wave. 943 00:52:21,000 --> 00:52:28,390 Because, for example, here, we have a different kx and kz, 944 00:52:28,390 --> 00:52:29,360 right? 945 00:52:29,360 --> 00:52:34,540 So what it means here, kx and ky-- 946 00:52:34,540 --> 00:52:35,540 well, what does it mean? 947 00:52:40,670 --> 00:52:42,920 Let's not go into Wigner distributions, right? 948 00:52:45,500 --> 00:52:49,160 OK, so let me say what I'm trying 949 00:52:49,160 --> 00:52:52,180 to say in plain language. 950 00:52:52,180 --> 00:52:56,290 The kx, ky, kz will define them for a plane wave. 951 00:52:56,290 --> 00:52:58,960 So a plane wave is this kind of thing, where you have 952 00:52:58,960 --> 00:53:01,020 a well-defined wave vector. 953 00:53:01,020 --> 00:53:05,158 And the entire wavefront is planar. 954 00:53:05,158 --> 00:53:06,700 Now, what you are saying is that if I 955 00:53:06,700 --> 00:53:11,480 have a spherical wavefront like this, locally, 956 00:53:11,480 --> 00:53:13,320 I can define normals, right? 957 00:53:13,320 --> 00:53:18,430 And I can call those wave vectors as well. 958 00:53:18,430 --> 00:53:20,140 And that is actually true. 959 00:53:20,140 --> 00:53:22,760 But they're not plane waves. 960 00:53:22,760 --> 00:53:24,800 You can think of them as rays. 961 00:53:24,800 --> 00:53:26,420 That's a good approximation. 962 00:53:26,420 --> 00:53:29,430 They would be the equivalent of geometrical rays, 963 00:53:29,430 --> 00:53:31,723 but not plane waves. 964 00:53:31,723 --> 00:53:33,140 But nevertheless, you are correct. 965 00:53:33,140 --> 00:53:35,210 They would-- these things are also paraxial. 966 00:53:35,210 --> 00:53:42,830 So they would also satisfy kx, ky much less than kz. 967 00:53:42,830 --> 00:53:43,380 That is true. 968 00:53:53,370 --> 00:53:54,576 Other questions? 969 00:54:02,520 --> 00:54:05,730 I'd like to talk about a fun topic. 970 00:54:05,730 --> 00:54:07,860 Today, I was-- I want to do electromagnetics. 971 00:54:07,860 --> 00:54:10,210 So I don't want too spend too much time on this. 972 00:54:10,210 --> 00:54:12,820 But I'll talk about it, because it's fun 973 00:54:12,820 --> 00:54:18,030 and very useful in many contexts. 974 00:54:18,030 --> 00:54:28,108 So in all of this business, we wrote the-- 975 00:54:28,108 --> 00:54:29,525 this is, again, the wave equation. 976 00:54:32,040 --> 00:54:37,420 But we didn't say anything about the speed of light. 977 00:54:37,420 --> 00:54:39,930 So actually, we did a little bit. 978 00:54:39,930 --> 00:54:43,680 I mentioned-- I guess this is better than a whiteboard. 979 00:54:43,680 --> 00:54:46,310 I have a history of what I've written throughout the lecture. 980 00:54:46,310 --> 00:54:49,470 Yeah, so at some point, we did mention this. 981 00:54:49,470 --> 00:54:52,890 We did mention that the index of refraction 982 00:54:52,890 --> 00:54:56,340 actually enters in this equation over here. 983 00:54:56,340 --> 00:55:02,130 But if you remember diligently from your geometrical optics, 984 00:55:02,130 --> 00:55:04,050 the index of refraction also happens 985 00:55:04,050 --> 00:55:05,833 to be a function of wavelength. 986 00:55:09,385 --> 00:55:10,510 It's not a constant, right? 987 00:55:10,510 --> 00:55:12,770 It is a function of wavelength. 988 00:55:12,770 --> 00:55:19,720 So what is means now is that in this equation over here, 989 00:55:19,720 --> 00:55:21,970 if I have a single frequency-- 990 00:55:21,970 --> 00:55:26,110 now, a single temporal frequency, a single omega-- 991 00:55:26,110 --> 00:55:28,390 then everything is fine. 992 00:55:28,390 --> 00:55:32,080 Because, sure, for this particular frequency, 993 00:55:32,080 --> 00:55:34,500 I can find the index of refraction. 994 00:55:34,500 --> 00:55:36,380 And I'm OK. 995 00:55:39,190 --> 00:55:46,950 However, the index of refraction is a function of wavelength. 996 00:55:46,950 --> 00:55:55,590 And the wavelength, yes, it is a function of frequency. 997 00:55:55,590 --> 00:56:00,040 We do know that this equation is true. 998 00:56:00,040 --> 00:56:04,280 But we also said, if you recall, or if you go back 999 00:56:04,280 --> 00:56:06,020 to the video of the first lecture 1000 00:56:06,020 --> 00:56:09,200 when I first presented this equation, I gave you a warning. 1001 00:56:09,200 --> 00:56:11,930 And I said that this is the dispersion relation. 1002 00:56:11,930 --> 00:56:16,310 But it is not the only possible dispersion relation. 1003 00:56:16,310 --> 00:56:18,670 It is possible to have waves where the dispersion 1004 00:56:18,670 --> 00:56:20,900 relation is different. 1005 00:56:20,900 --> 00:56:23,690 OK, first of all, just to make sure 1006 00:56:23,690 --> 00:56:27,260 that the [INAUDIBLE] stuff about dispersion that we discussed 1007 00:56:27,260 --> 00:56:29,430 about the glass, it is true. 1008 00:56:29,430 --> 00:56:32,630 In other words, in glass, yes, the index of refraction 1009 00:56:32,630 --> 00:56:35,310 is a function of wavelength. 1010 00:56:35,310 --> 00:56:38,010 So this is a plot that we saw before. 1011 00:56:38,010 --> 00:56:40,190 But now, in addition to that, I'm 1012 00:56:40,190 --> 00:56:43,670 talking about a slightly different type of dispersion. 1013 00:56:43,670 --> 00:56:48,770 So this dispersion happens, for example, in waveguides. 1014 00:56:48,770 --> 00:56:51,470 In everything that we'll do so far, 1015 00:56:51,470 --> 00:56:54,390 we sort of assume that light propagates freely. 1016 00:56:54,390 --> 00:57:00,230 We may put some lenses or whatever, 1017 00:57:00,230 --> 00:57:02,480 prisms in the path of the light, that 1018 00:57:02,480 --> 00:57:05,630 is refractors, and mirrors, and so on. 1019 00:57:05,630 --> 00:57:07,370 But all of these are relatively large. 1020 00:57:07,370 --> 00:57:11,850 We never try to confine the light to a very small space. 1021 00:57:11,850 --> 00:57:12,880 But you can do that. 1022 00:57:12,880 --> 00:57:16,760 You can create waveguides. 1023 00:57:16,760 --> 00:57:20,570 Actually, for visible light, you will never actually use a metal 1024 00:57:20,570 --> 00:57:21,540 here. 1025 00:57:21,540 --> 00:57:23,570 You would use a dielectric waveguide. 1026 00:57:23,570 --> 00:57:26,430 But in microwaves, this is done very commonly. 1027 00:57:26,430 --> 00:57:33,020 In microwaves, people trap the wave in a metal tube 1028 00:57:33,020 --> 00:57:38,660 as if it were water, I suppose, or some kind of a liquid. 1029 00:57:38,660 --> 00:57:41,160 You just have a hollow tube that is a metal. 1030 00:57:41,160 --> 00:57:44,220 And then you launch the microwave inside the tube. 1031 00:57:44,220 --> 00:57:47,530 And to give you an idea, the wavelength in microwaves 1032 00:57:47,530 --> 00:57:49,040 is in the order of-- 1033 00:57:49,040 --> 00:57:56,150 well, as the name suggests, it is the order of-- 1034 00:57:56,150 --> 00:57:58,370 the frequency's gigahertz. 1035 00:57:58,370 --> 00:58:01,340 So the wavelength should be between millimeters 1036 00:58:01,340 --> 00:58:02,060 and centimeters. 1037 00:58:02,060 --> 00:58:07,455 That is typically-- actually, less, 1038 00:58:07,455 --> 00:58:09,080 no the wavelength is a little bit less, 1039 00:58:09,080 --> 00:58:10,870 maybe in the order of millimeters, 1040 00:58:10,870 --> 00:58:13,820 typical millimeter. 1041 00:58:13,820 --> 00:58:16,098 So the size of these tubes might also 1042 00:58:16,098 --> 00:58:17,640 be on the order of a few millimeters. 1043 00:58:17,640 --> 00:58:21,200 So now you have a guide that is approximately 1044 00:58:21,200 --> 00:58:25,040 of size of a few wavelengths. 1045 00:58:25,040 --> 00:58:29,140 So in that case, it turns this equation, c equals lambda nu, 1046 00:58:29,140 --> 00:58:30,790 does not hold anymore. 1047 00:58:30,790 --> 00:58:32,740 It is true for a free space. 1048 00:58:32,740 --> 00:58:35,080 But it is not true in a waveguide. 1049 00:58:35,080 --> 00:58:37,480 The reason it is not true in a waveguide 1050 00:58:37,480 --> 00:58:40,850 is because the wave must satisfy boundary conditions. 1051 00:58:40,850 --> 00:58:43,020 If you recall when we gave the wave equation, 1052 00:58:43,020 --> 00:58:45,490 we said we would need to satisfy an initial condition 1053 00:58:45,490 --> 00:58:47,070 and boundary conditions. 1054 00:58:47,070 --> 00:58:49,595 In free space, there is no boundary condition. 1055 00:58:49,595 --> 00:58:51,220 Actually, there is a boundary condition 1056 00:58:51,220 --> 00:58:53,410 that the wave must vanish with infinity. 1057 00:58:53,410 --> 00:58:56,710 OK, fine, but that doesn't have any implication 1058 00:58:56,710 --> 00:58:59,860 in our calculations. 1059 00:58:59,860 --> 00:59:02,140 But, for example, in the case of a metal, 1060 00:59:02,140 --> 00:59:06,790 you have to force the wave to be 0 on the metal. 1061 00:59:06,790 --> 00:59:09,940 Because, well, we'll see later in electromagnetics 1062 00:59:09,940 --> 00:59:10,990 why this is the case. 1063 00:59:10,990 --> 00:59:13,300 But take my word for it. 1064 00:59:16,920 --> 00:59:20,250 On the metal, the field must equals 0. 1065 00:59:20,250 --> 00:59:24,250 So now if you have a sinusoid in this metal, 1066 00:59:24,250 --> 00:59:26,670 the sinusoid is in kind of a straitjacket. 1067 00:59:26,670 --> 00:59:30,480 Because it is sinusoidally varying. 1068 00:59:30,480 --> 00:59:34,110 But also, it must be over such a period 1069 00:59:34,110 --> 00:59:38,650 that it vanishes at the edge of the waveguide. 1070 00:59:38,650 --> 00:59:40,290 So, for example, it can be-- 1071 00:59:44,420 --> 00:59:49,530 if this is the waveguide, the sinusoid can be like this. 1072 00:59:49,530 --> 00:59:51,500 That's fine, because it vanishes. 1073 00:59:51,500 --> 00:59:52,830 It can be like this. 1074 00:59:52,830 --> 00:59:53,920 That's also fine. 1075 00:59:53,920 --> 00:59:56,860 It can be like this, and so on. 1076 00:59:56,860 --> 01:00:05,820 But, for example, can it be like this? 1077 01:00:05,820 --> 01:00:08,620 No, that wave cannot. 1078 01:00:08,620 --> 01:00:14,140 OK, so I will not do the full derivation. 1079 01:00:14,140 --> 01:00:16,860 And again, it's a little bit-- 1080 01:00:16,860 --> 01:00:19,770 well, we would have to solve the partial differential 1081 01:00:19,770 --> 01:00:21,760 equation in a serious way. 1082 01:00:21,760 --> 01:00:24,000 But it turns out that this straitjacket 1083 01:00:24,000 --> 01:00:26,880 that we put the light in actually 1084 01:00:26,880 --> 01:00:28,770 results in a different dispersion 1085 01:00:28,770 --> 01:00:30,030 relation that looks like this. 1086 01:00:30,030 --> 01:00:32,790 It is not the c equals lambda nu anymore, but it's this one. 1087 01:00:32,790 --> 01:00:34,500 AUDIENCE: I have a question? 1088 01:00:34,500 --> 01:00:35,780 GEORGE BARBASTATHIS: Yes? 1089 01:00:35,780 --> 01:00:39,050 AUDIENCE: What happens if a is smaller than lambda? 1090 01:00:42,930 --> 01:00:44,930 GEORGE BARBASTATHIS: Then the wave will not even 1091 01:00:44,930 --> 01:00:46,190 get into the waveguide. 1092 01:00:46,190 --> 01:00:47,510 It will become evanescent. 1093 01:00:47,510 --> 01:00:48,490 So it will not go in. 1094 01:00:51,609 --> 01:00:52,442 AUDIENCE: Thank you. 1095 01:00:56,900 --> 01:00:58,690 AUDIENCE: What is shown in the diagram, 1096 01:00:58,690 --> 01:01:00,800 isn't it a standing wave? 1097 01:01:00,800 --> 01:01:03,530 Because it's not-- oscillation is-- 1098 01:01:06,940 --> 01:01:07,940 the way you have drawn-- 1099 01:01:07,940 --> 01:01:09,940 GEORGE BARBASTATHIS: Yeah, it is a standing wave 1100 01:01:09,940 --> 01:01:11,480 in the vertical direction. 1101 01:01:11,480 --> 01:01:13,105 AUDIENCE: Yeah, but how does it propa-- 1102 01:01:13,105 --> 01:01:15,355 GEORGE BARBASTATHIS: Well, if the [INAUDIBLE] of it is 1103 01:01:15,355 --> 01:01:16,490 bouncing back and forth-- 1104 01:01:16,490 --> 01:01:18,032 another way to think of the waveguide 1105 01:01:18,032 --> 01:01:20,970 is that the light is-- 1106 01:01:20,970 --> 01:01:23,100 you can think, actually, in terms of rays. 1107 01:01:23,100 --> 01:01:26,950 You can think of rays bouncing back and forth. 1108 01:01:26,950 --> 01:01:29,635 But you can also have the conjugate ray. 1109 01:01:29,635 --> 01:01:30,635 These are both possible. 1110 01:01:34,230 --> 01:01:37,230 And the result is a standing wave in this sentence. 1111 01:01:37,230 --> 01:01:38,760 But you can still have propagation 1112 01:01:38,760 --> 01:01:39,780 on the horizontal axis. 1113 01:01:39,780 --> 01:01:43,110 Because there is a component that goes towards the right. 1114 01:01:47,710 --> 01:01:54,850 OK, so when you get an equation like this one which 1115 01:01:54,850 --> 01:01:58,060 relates omega to k-- and, of course, as you can imagine, 1116 01:01:58,060 --> 01:02:00,820 now there's a great variety of such equations 1117 01:02:00,820 --> 01:02:01,780 that you could have. 1118 01:02:01,780 --> 01:02:03,490 This is the equation that happens to be 1119 01:02:03,490 --> 01:02:05,470 true for the metal waveguide. 1120 01:02:05,470 --> 01:02:07,840 If you replace it with a different waveguide, 1121 01:02:07,840 --> 01:02:10,050 for example, if you have dielectrics 1122 01:02:10,050 --> 01:02:14,530 of different indices here, then this equation changes. 1123 01:02:14,530 --> 01:02:16,840 The bottom line is that if you have this equation, then 1124 01:02:16,840 --> 01:02:18,630 you can plot one of the two. 1125 01:02:18,630 --> 01:02:22,990 You have one term here that is determined by geometry. 1126 01:02:22,990 --> 01:02:25,540 a is the size of the guide. 1127 01:02:25,540 --> 01:02:29,650 And then you have the wave vector and the frequency. 1128 01:02:29,650 --> 01:02:32,410 And, of course, you have the free velocity of light. 1129 01:02:32,410 --> 01:02:35,260 OK, omega is fundamental. 1130 01:02:35,260 --> 01:02:37,100 As soon as you specify the frequency 1131 01:02:37,100 --> 01:02:39,100 of the electromagnetic wave, nothing 1132 01:02:39,100 --> 01:02:42,295 can change that anymore since we're in the linear region. 1133 01:02:42,295 --> 01:02:44,170 But the question is, what is the wave vector? 1134 01:02:44,170 --> 01:02:45,730 So the wave vector is really what 1135 01:02:45,730 --> 01:02:48,620 you are after in this equation. 1136 01:02:48,620 --> 01:02:49,810 Omega is given. 1137 01:02:49,810 --> 01:02:52,660 It is, say, 10 to the 15 Hertz. 1138 01:02:52,660 --> 01:02:56,040 What is the k? 1139 01:02:56,040 --> 01:02:58,540 Or in other words, what is the wavelength? 1140 01:02:58,540 --> 01:03:00,390 Now, for reasons which will become apparent 1141 01:03:00,390 --> 01:03:05,142 in a moment, the people who invented these diagrams, 1142 01:03:05,142 --> 01:03:07,350 they actually plot them kind of the other way around. 1143 01:03:07,350 --> 01:03:10,200 Instead of omega on the horizontal axis, 1144 01:03:10,200 --> 01:03:13,990 since omega is given, they put omega on the vertical axis. 1145 01:03:13,990 --> 01:03:16,330 And the horizontal axis is k. 1146 01:03:16,330 --> 01:03:18,330 So the way you read this diagram is-- of course, 1147 01:03:18,330 --> 01:03:20,380 you solve this equation. 1148 01:03:20,380 --> 01:03:22,590 And the way you read it is that if I give you 1149 01:03:22,590 --> 01:03:26,810 a certain omega, then you have to draw a horizontal line. 1150 01:03:26,810 --> 01:03:32,550 And wherever this horizontal line meets the plot, this-- 1151 01:03:32,550 --> 01:03:35,312 you read the abscissa and-- 1152 01:03:35,312 --> 01:03:36,270 I never get this right. 1153 01:03:36,270 --> 01:03:38,380 Is this the abscissa or the other one? 1154 01:03:38,380 --> 01:03:39,213 That's the abscissa? 1155 01:03:39,213 --> 01:03:40,432 AUDIENCE: [INAUDIBLE] 1156 01:03:40,432 --> 01:03:42,140 GEORGE BARBASTATHIS: That's the ordinate? 1157 01:03:42,140 --> 01:03:43,788 OK. 1158 01:03:43,788 --> 01:03:45,580 I must come up with a mnemonic to remember. 1159 01:03:45,580 --> 01:03:49,030 Anyway, so the value of the horizontal axis 1160 01:03:49,030 --> 01:03:52,660 gives you the wave vector, and hence, the wavelength. 1161 01:03:52,660 --> 01:03:55,908 Now, interestingly, if you were in free space, 1162 01:03:55,908 --> 01:03:57,450 this would look like a straight line. 1163 01:03:57,450 --> 01:04:00,420 Because, well, of course in free space-- 1164 01:04:00,420 --> 01:04:02,400 I have written this equation many times. 1165 01:04:02,400 --> 01:04:04,500 I don't feel like writing it again. 1166 01:04:04,500 --> 01:04:07,930 In free space-- well, let me do it once more. 1167 01:04:07,930 --> 01:04:18,480 So in free space, I have that c equals k over 2 pi. 1168 01:04:18,480 --> 01:04:20,475 That's also known as lambda times-- 1169 01:04:24,070 --> 01:04:25,920 what am I doing? 1170 01:04:25,920 --> 01:04:26,940 The other way around-- 1171 01:04:26,940 --> 01:04:30,720 2 pi over k, that is also known as lambda, 1172 01:04:30,720 --> 01:04:33,460 times omega over 2 pi. 1173 01:04:33,460 --> 01:04:38,700 So in free space, then, you have omega equals ck. 1174 01:04:38,700 --> 01:04:44,590 This is an equivalent way of writing this equation here. 1175 01:04:44,590 --> 01:04:46,950 These are basically the same equation. 1176 01:04:46,950 --> 01:04:50,520 So in free space, this is a straight line. 1177 01:04:54,770 --> 01:04:57,200 Now, if you have an equation like this one, 1178 01:04:57,200 --> 01:05:03,010 you can see that at relatively high frequencies, 1179 01:05:03,010 --> 01:05:04,030 they approach. 1180 01:05:04,030 --> 01:05:07,100 So, indeed, all of the-- oh, by the way, what are these curves? 1181 01:05:07,100 --> 01:05:10,150 So this equation involves an integral parameter. 1182 01:05:10,150 --> 01:05:12,760 You can plug in-- 1183 01:05:12,760 --> 01:05:16,112 well, not quite 0, please erase that from your notes. 1184 01:05:16,112 --> 01:05:17,320 I should not have put 0 here. 1185 01:05:17,320 --> 01:05:18,550 0 doesn't make sense. 1186 01:05:18,550 --> 01:05:22,330 But you can have plus/minus 1, plus/minus 2, and so on. 1187 01:05:22,330 --> 01:05:24,310 And these are known as modes. 1188 01:05:24,310 --> 01:05:27,970 So the index defines the mode that 1189 01:05:27,970 --> 01:05:31,810 is launched into the waveguide. 1190 01:05:31,810 --> 01:05:38,030 So each one of these modes, at very high frequencies, 1191 01:05:38,030 --> 01:05:41,690 it actually kind of approaches the free space mode. 1192 01:05:41,690 --> 01:05:44,030 But as you go to lower frequencies, 1193 01:05:44,030 --> 01:05:47,120 you see that it bends away and then eventually hits 1194 01:05:47,120 --> 01:05:49,900 the ordinate axis. 1195 01:05:49,900 --> 01:05:51,620 And there's a gap here now. 1196 01:05:51,620 --> 01:05:57,600 Because there's nothing in this place over here. 1197 01:05:57,600 --> 01:05:59,400 So what will happen if you come up, 1198 01:05:59,400 --> 01:06:03,200 if you use an omega that falls in this range? 1199 01:06:09,520 --> 01:06:10,660 AUDIENCE: Evanescent wave. 1200 01:06:10,660 --> 01:06:12,660 GEORGE BARBASTATHIS: Correct-- actually, someone 1201 01:06:12,660 --> 01:06:13,830 asked it before in Boston. 1202 01:06:13,830 --> 01:06:16,050 Someone asked me before, what happens 1203 01:06:16,050 --> 01:06:19,530 if you pick a to be smaller than a wavelength? 1204 01:06:19,530 --> 01:06:21,940 Well, this is what happens. 1205 01:06:21,940 --> 01:06:25,250 In fact, I wouldn't espouse it exactly like this. 1206 01:06:25,250 --> 01:06:27,390 A more accurate way to say it is what 1207 01:06:27,390 --> 01:06:31,560 happens if you pick an omega which is very small? 1208 01:06:31,560 --> 01:06:36,060 So therefore, given the value of a and given the value of omega, 1209 01:06:36,060 --> 01:06:37,860 you cannot find the k anymore. 1210 01:06:37,860 --> 01:06:40,130 The square root becomes imaginary. 1211 01:06:40,130 --> 01:06:42,950 Well, if that happens, the wave does not enter the waveguide. 1212 01:06:42,950 --> 01:06:44,730 It becomes evanescent, as we say. 1213 01:06:44,730 --> 01:06:46,800 So basically, it may penetrate a little bit 1214 01:06:46,800 --> 01:06:48,300 for a couple of wavelengths. 1215 01:06:48,300 --> 01:06:51,930 And then it will exponentially decay. 1216 01:06:51,930 --> 01:06:55,500 Now, if you-- as you go progressively-- so, 1217 01:06:55,500 --> 01:06:57,600 if you launch at very low frequency, 1218 01:06:57,600 --> 01:06:59,800 the wave does not enter. 1219 01:06:59,800 --> 01:07:03,360 As you start increasing the frequency now, 1220 01:07:03,360 --> 01:07:06,540 you will see that the intersection only 1221 01:07:06,540 --> 01:07:07,823 meets one mode. 1222 01:07:07,823 --> 01:07:09,615 So basically, at the range of frequencies-- 1223 01:07:09,615 --> 01:07:10,870 I don't know what it is here. 1224 01:07:10,870 --> 01:07:13,230 This looks like 3 or something. 1225 01:07:13,230 --> 01:07:16,290 So between-- so the way you read this axis 1226 01:07:16,290 --> 01:07:18,480 is you read the ordinate. 1227 01:07:18,480 --> 01:07:21,570 And you multiply by three times 10 to the 14. 1228 01:07:21,570 --> 01:07:26,970 So, for example, 5 actually means 15 times 10 1229 01:07:26,970 --> 01:07:28,620 to the 14 Hertz. 1230 01:07:28,620 --> 01:07:32,130 OK, so then between 3, the value f3, which 1231 01:07:32,130 --> 01:07:38,400 is 9 times 10 to the 14, and the value, whatever it is, 7, 1232 01:07:38,400 --> 01:07:40,830 you only launch one mode. 1233 01:07:40,830 --> 01:07:44,180 As you go up in frequency, if your intersection here 1234 01:07:44,180 --> 01:07:46,800 meets more than one line, then actually you 1235 01:07:46,800 --> 01:07:51,050 can have multiple modes in this guide. 1236 01:07:51,050 --> 01:07:54,260 Anyway, so these modes, these little wiggles 1237 01:07:54,260 --> 01:07:56,940 that I drew over here, the first mode 1238 01:07:56,940 --> 01:08:00,480 is the field shape that looks like this, only half 1239 01:08:00,480 --> 01:08:02,360 of the period of the sinusoid. 1240 01:08:02,360 --> 01:08:07,050 The next mode, you allow one full period. 1241 01:08:07,050 --> 01:08:09,330 The next mode, you allow 1 1/2 period, 1242 01:08:09,330 --> 01:08:10,770 then so on and so forth. 1243 01:08:10,770 --> 01:08:12,750 So as you go higher in frequency, 1244 01:08:12,750 --> 01:08:14,370 it means that the wave-- 1245 01:08:14,370 --> 01:08:17,649 basically, what it means is that the waveguide now-- 1246 01:08:17,649 --> 01:08:20,399 or, I should say the wavelength of the wave 1247 01:08:20,399 --> 01:08:24,540 becomes progressively smaller than the size of the waveguide. 1248 01:08:24,540 --> 01:08:29,649 So therefore, you can pack more modes into the guide. 1249 01:08:29,649 --> 01:08:31,123 So I only plotted three here. 1250 01:08:31,123 --> 01:08:32,290 Of course, there's infinite. 1251 01:08:32,290 --> 01:08:34,240 But anyway, as you go up in frequency, 1252 01:08:34,240 --> 01:08:37,270 you have the possibility to excite more than one mode. 1253 01:08:37,270 --> 01:08:38,743 Now, how do you excite the modes? 1254 01:08:38,743 --> 01:08:41,410 Now, that depends, of course, on your initial conditions, right? 1255 01:08:41,410 --> 01:08:44,080 If you create a distribution in the entrance 1256 01:08:44,080 --> 01:08:46,120 of the waveguide that looks like this, 1257 01:08:46,120 --> 01:08:47,620 then you will excite the first mode. 1258 01:08:47,620 --> 01:08:50,200 If you create a distribution that looks like this, 1259 01:08:50,200 --> 01:08:52,359 well, if the second mode is allowable, 1260 01:08:52,359 --> 01:08:55,029 that is if you are above the green line, 1261 01:08:55,029 --> 01:08:56,410 then you will excite it. 1262 01:08:56,410 --> 01:08:57,939 If you are below the green line, you 1263 01:08:57,939 --> 01:08:59,350 will probably excite nothing. 1264 01:08:59,350 --> 01:09:01,880 Because the first mode cannot be excited this way. 1265 01:09:05,727 --> 01:09:07,310 The reason I'm telling you this is not 1266 01:09:07,310 --> 01:09:10,069 because I want to do waveguide theory, but because I 1267 01:09:10,069 --> 01:09:12,920 but because I want to do something that's called a beat. 1268 01:09:12,920 --> 01:09:15,740 And a beat is an extremely useful concept, 1269 01:09:15,740 --> 01:09:19,770 both in the space domain and in the time domain. 1270 01:09:19,770 --> 01:09:22,189 So in the time domain, the beat is actually-- 1271 01:09:22,189 --> 01:09:23,390 it is a very simple thing. 1272 01:09:23,390 --> 01:09:25,460 It is defined as the superposition 1273 01:09:25,460 --> 01:09:30,390 of two sinusoids of slightly different frequency. 1274 01:09:30,390 --> 01:09:33,140 So we said this many times, that if the wave is linear, 1275 01:09:33,140 --> 01:09:35,569 therefore a superposition is allowed. 1276 01:09:35,569 --> 01:09:38,720 If I have two solutions to the wave equation. 1277 01:09:38,720 --> 01:09:39,560 I take the sum. 1278 01:09:39,560 --> 01:09:41,819 It is still a solution. 1279 01:09:41,819 --> 01:09:43,979 So now what you can see here very easily-- 1280 01:09:43,979 --> 01:09:46,960 that's a diagram that I stole from the textbook. 1281 01:09:46,960 --> 01:09:50,779 Because the two waves have slightly different frequency, 1282 01:09:50,779 --> 01:09:51,890 they will misalign. 1283 01:09:51,890 --> 01:09:55,400 So you can, for example, start at the point 1284 01:09:55,400 --> 01:09:58,172 where the waves are exactly aligned. 1285 01:09:58,172 --> 01:09:59,630 So they kind of oscillate together. 1286 01:09:59,630 --> 01:10:02,150 When one is positive, the other is positive. 1287 01:10:02,150 --> 01:10:03,860 But because the period is different, 1288 01:10:03,860 --> 01:10:06,350 or the frequency is different, a little bit later, 1289 01:10:06,350 --> 01:10:09,410 they will kind of separate. 1290 01:10:09,410 --> 01:10:12,080 And now, over here, you can see that this is the opposite. 1291 01:10:12,080 --> 01:10:15,390 When one of them is positive, the other is negative. 1292 01:10:15,390 --> 01:10:17,750 And if you also happen to judiciously pick 1293 01:10:17,750 --> 01:10:19,790 them to have the same amplitude, it 1294 01:10:19,790 --> 01:10:24,240 means that here when, as we say, they oscillate out of phase, 1295 01:10:24,240 --> 01:10:30,210 it means that the wave amplitude overall becomes very small. 1296 01:10:30,210 --> 01:10:33,480 And so-- and of course, in the other case, 1297 01:10:33,480 --> 01:10:35,340 where they are in phase, the amplitude 1298 01:10:35,340 --> 01:10:38,010 becomes twice the amplitude of one. 1299 01:10:38,010 --> 01:10:41,190 So this is sort of a static picture from the book. 1300 01:10:41,190 --> 01:10:44,310 I also made a movie out of it. 1301 01:10:44,310 --> 01:10:46,534 So here you see the beat propagating. 1302 01:10:50,800 --> 01:10:55,595 OK, so now, what is really interesting about this-- 1303 01:10:55,595 --> 01:10:56,720 I'm going to play it again. 1304 01:11:04,390 --> 01:11:06,370 If you look at it carefully-- 1305 01:11:06,370 --> 01:11:09,720 it takes a-- let me play it once more. 1306 01:11:09,720 --> 01:11:10,970 Actually, I think I can just-- 1307 01:11:19,790 --> 01:11:22,900 if you look at it carefully, you will 1308 01:11:22,900 --> 01:11:27,660 see that there is two different velocities here. 1309 01:11:30,072 --> 01:11:31,280 There's two things happening. 1310 01:11:31,280 --> 01:11:37,120 One is you have this envelope that is moving to the right. 1311 01:11:37,120 --> 01:11:39,460 But also, inside the envelope, you 1312 01:11:39,460 --> 01:11:44,220 have small fringes, wiggles that are also moving to the right. 1313 01:11:44,220 --> 01:11:48,305 Now, I told you, of course, but if you look at it once 1314 01:11:48,305 --> 01:11:50,430 again more carefully, you will see that the two are 1315 01:11:50,430 --> 01:11:52,050 moving with different speeds. 1316 01:11:56,610 --> 01:11:57,840 Can you see it? 1317 01:12:00,750 --> 01:12:03,307 The wiggles are moving faster than the envelope. 1318 01:12:03,307 --> 01:12:05,390 And you can see that the wiggles are moving faster 1319 01:12:05,390 --> 01:12:07,830 than the envelop if you look here 1320 01:12:07,830 --> 01:12:10,095 near the vicinity of the null. 1321 01:12:10,095 --> 01:12:11,970 You will see that there's a little bit of a-- 1322 01:12:11,970 --> 01:12:14,820 once in a while, there's a pulse, like a little pulse. 1323 01:12:14,820 --> 01:12:18,390 That means that the fringe are passing by. 1324 01:12:18,390 --> 01:12:20,010 So the reason this is happening-- 1325 01:12:20,010 --> 01:12:22,110 and now I will not derive it on the paper. 1326 01:12:22,110 --> 01:12:25,030 Because it will end a little bit late. 1327 01:12:25,030 --> 01:12:28,200 But the reason this is happening is 1328 01:12:28,200 --> 01:12:30,460 because if you take the superposition of the two 1329 01:12:30,460 --> 01:12:34,170 sinusoids, now we have to be a little bit careful. 1330 01:12:34,170 --> 01:12:37,180 Because they have different frequencies. 1331 01:12:37,180 --> 01:12:39,270 So if we use phasor-- we could use phasors. 1332 01:12:39,270 --> 01:12:42,270 But we might get a little bit confused. 1333 01:12:42,270 --> 01:12:45,030 So for that reason, I just wrote it in the real space 1334 01:12:45,030 --> 01:12:46,363 simply with cosines. 1335 01:12:46,363 --> 01:12:48,030 So you do the same thing with it before. 1336 01:12:48,030 --> 01:12:50,260 You take the sum of the two cosines. 1337 01:12:50,260 --> 01:12:52,170 You apply the trigonometric identity 1338 01:12:52,170 --> 01:12:53,910 to write it as a product. 1339 01:12:53,910 --> 01:12:57,000 And then you find that the product contains two terms. 1340 01:12:57,000 --> 01:13:01,940 One of them-- actually, by the way, the-- 1341 01:13:01,940 --> 01:13:02,630 oh, shoot. 1342 01:13:02,630 --> 01:13:04,770 There's one more error in the notes. 1343 01:13:04,770 --> 01:13:07,800 There should be a cosine between the two parentheses. 1344 01:13:07,800 --> 01:13:10,200 There shouldn't be a cosine. 1345 01:13:10,200 --> 01:13:12,286 I better write down the correct expression. 1346 01:13:37,630 --> 01:13:47,210 OK, where kc equals k1 plus k2 over 2, 1347 01:13:47,210 --> 01:13:57,340 and the same for omega c, and km equals k1 minus k2 over 2, 1348 01:13:57,340 --> 01:13:58,590 so one of them is the average. 1349 01:13:58,590 --> 01:14:01,130 The other is the difference. 1350 01:14:08,390 --> 01:14:12,220 So now each one of those, each one of these cosines-- 1351 01:14:12,220 --> 01:14:13,680 OK, there's a missing cosine here. 1352 01:14:13,680 --> 01:14:15,437 But imagine that there was a cosine. 1353 01:14:15,437 --> 01:14:17,020 Each one of those is a traveling wave. 1354 01:14:20,010 --> 01:14:22,680 So basically, what you see now is the product of the two 1355 01:14:22,680 --> 01:14:23,460 traveling waves. 1356 01:14:23,460 --> 01:14:26,430 And you can see very clearly it's a product here. 1357 01:14:26,430 --> 01:14:32,920 This is-- I have one sinusoid of relatively low spatial period 1358 01:14:32,920 --> 01:14:38,400 and another sinusoid of relatively fast spatial period. 1359 01:14:38,400 --> 01:14:39,562 And I multiply them. 1360 01:14:39,562 --> 01:14:40,770 But they're both propagating. 1361 01:14:40,770 --> 01:14:44,310 Because they're both of the form kz minus omega t. 1362 01:14:44,310 --> 01:14:47,220 But they're propagating with different speeds. 1363 01:14:47,220 --> 01:14:49,970 Because the speed of the-- 1364 01:14:49,970 --> 01:14:53,070 oh, and-- so, what are the names? 1365 01:14:53,070 --> 01:14:58,830 Well, the fast sinusoid here is called the carrier. 1366 01:14:58,830 --> 01:15:00,562 Those of you who are electrical engineers 1367 01:15:00,562 --> 01:15:01,770 are very familiar with those. 1368 01:15:01,770 --> 01:15:03,990 The carrier wave is-- 1369 01:15:03,990 --> 01:15:09,450 in radiowaves, for example, it is the frequency upon which 1370 01:15:09,450 --> 01:15:11,910 we'll modulate the signal. 1371 01:15:11,910 --> 01:15:20,630 And the slow sinusoid, that's called the modulation. 1372 01:15:20,630 --> 01:15:27,370 And-- I lost my train of thought. 1373 01:15:27,370 --> 01:15:31,270 All right, but each one of those is a propagating wave. 1374 01:15:31,270 --> 01:15:36,292 So therefore, their velocities would be equal to the-- 1375 01:15:36,292 --> 01:15:38,250 it will depend on which wave I'm talking about. 1376 01:15:38,250 --> 01:15:40,960 So, for example, the velocity of the carrier 1377 01:15:40,960 --> 01:15:44,290 will equal omega of the carrier divided 1378 01:15:44,290 --> 01:15:46,270 by the k of the carrier. 1379 01:15:46,270 --> 01:15:48,070 And the velocity of the modulation 1380 01:15:48,070 --> 01:15:50,110 will equal omega of the modulation 1381 01:15:50,110 --> 01:15:52,450 over k of the modulation. 1382 01:15:52,450 --> 01:15:55,360 OK, and so this is not magic. 1383 01:15:55,360 --> 01:15:57,430 We'll just apply the theory that we learned. 1384 01:15:57,430 --> 01:16:00,280 But now it is the two waves that are propagating. 1385 01:16:00,280 --> 01:16:03,670 And, of course, in general, the two velocities are different. 1386 01:16:03,670 --> 01:16:06,340 Because the two rations over here, 1387 01:16:06,340 --> 01:16:08,620 they have no reason to be the same. 1388 01:16:08,620 --> 01:16:10,240 In fact, it is quite the opposite. 1389 01:16:10,240 --> 01:16:12,350 Most of the time they are different. 1390 01:16:15,870 --> 01:16:21,230 So this velocity, I actually used a slightly different 1391 01:16:21,230 --> 01:16:22,220 notation. 1392 01:16:22,220 --> 01:16:28,570 The velocity of the carrier is called the phase velocity. 1393 01:16:28,570 --> 01:16:30,280 And the velocity of the modulation 1394 01:16:30,280 --> 01:16:32,037 is called the group velocity. 1395 01:16:50,940 --> 01:16:55,280 So the group velocity, it equals, as you can see, 1396 01:16:55,280 --> 01:16:59,260 the ratio of the difference of omega 1397 01:16:59,260 --> 01:17:02,990 to the ratio of the difference of k. 1398 01:17:02,990 --> 01:17:06,983 So if you imagine a beat-- 1399 01:17:06,983 --> 01:17:08,400 this is called a beat, by the way. 1400 01:17:08,400 --> 01:17:09,450 I think I said it before. 1401 01:17:09,450 --> 01:17:12,900 So if you imagine a beat where the two frequencies of the two 1402 01:17:12,900 --> 01:17:16,463 numbers are very close, then it will become the derivative. 1403 01:17:21,770 --> 01:17:24,590 So the group velocity is basically 1404 01:17:24,590 --> 01:17:26,630 the derivative of the-- 1405 01:17:26,630 --> 01:17:29,960 whatever omega is as a function of k, 1406 01:17:29,960 --> 01:17:32,450 if you take the derivative of that expression, 1407 01:17:32,450 --> 01:17:33,980 you get the group velocity. 1408 01:17:33,980 --> 01:17:37,070 The phase velocity is the same velocity 1409 01:17:37,070 --> 01:17:40,020 we've been talking about so far while we're doing waves. 1410 01:17:40,020 --> 01:17:43,040 So there's nothing new about the phase velocity. 1411 01:17:43,040 --> 01:17:47,480 It is still the ratio of omega over k. 1412 01:17:47,480 --> 01:17:50,060 But the group velocity is the derivative. 1413 01:17:50,060 --> 01:17:51,680 And the reason now-- 1414 01:17:51,680 --> 01:17:53,460 ah, now the reason is revealed. 1415 01:17:53,460 --> 01:17:54,580 AUDIENCE: George? 1416 01:17:54,580 --> 01:17:56,247 GEORGE BARBASTATHIS: There's a question? 1417 01:17:56,247 --> 01:17:58,470 AUDIENCE: Yeah, why did the derivative come about? 1418 01:17:58,470 --> 01:18:00,350 GEORGE BARBASTATHIS: Yeah, you're right. 1419 01:18:00,350 --> 01:18:01,740 I kind of blasted through. 1420 01:18:01,740 --> 01:18:04,300 So if you look at the group velocity, 1421 01:18:04,300 --> 01:18:07,760 it equals omega of the modulation 1422 01:18:07,760 --> 01:18:09,720 over k of the modulation. 1423 01:18:09,720 --> 01:18:13,120 And this equals omega 1 minus omega 2 over 2 1424 01:18:13,120 --> 01:18:16,250 over k1 minus k2 over 2. 1425 01:18:16,250 --> 01:18:20,960 That is omega 1 minus omega 2 over k1 minus k2. 1426 01:18:20,960 --> 01:18:23,220 And if you take omega one very close 1427 01:18:23,220 --> 01:18:26,340 but not quite the same and k1 very close 1428 01:18:26,340 --> 01:18:30,167 but not quite the same, then v sub g will become d omega dk. 1429 01:18:33,510 --> 01:18:35,420 So that is the reason people put omega 1430 01:18:35,420 --> 01:18:37,440 on the vertical axis of it here. 1431 01:18:37,440 --> 01:18:39,060 Because the group velocity, then, 1432 01:18:39,060 --> 01:18:44,900 is simply obtained as the slope of the dispersion diagram. 1433 01:18:44,900 --> 01:18:48,200 And that's fascinating, if you think about it. 1434 01:18:48,200 --> 01:18:51,470 Because it suggests some weird things. 1435 01:18:51,470 --> 01:19:05,090 For example, it suggests that if I could somehow 1436 01:19:05,090 --> 01:19:08,780 create this pendulum diagram that 1437 01:19:08,780 --> 01:19:12,490 looks like this, what is the strange thing about this? 1438 01:19:16,520 --> 01:19:19,520 That's right, it has a negative group philosophy, 1439 01:19:19,520 --> 01:19:24,800 which means that in the wave, the carrier and the modulation, 1440 01:19:24,800 --> 01:19:26,830 they go in opposite directions. 1441 01:19:26,830 --> 01:19:28,850 The phase velocity is going to the left. 1442 01:19:28,850 --> 01:19:30,600 The group velocity is going to the right. 1443 01:19:30,600 --> 01:19:32,433 I meant to make a movie of this to show you. 1444 01:19:32,433 --> 01:19:34,230 It actually looks a little bit weird. 1445 01:19:34,230 --> 01:19:36,170 But I didn't have time to do it. 1446 01:19:36,170 --> 01:19:37,750 And I'll show you one next time. 1447 01:19:37,750 --> 01:19:42,200 But now, of course, it demands-- it's interesting. 1448 01:19:42,200 --> 01:19:43,960 How can you do something like this? 1449 01:19:43,960 --> 01:19:49,390 It turns out to be an extremely hot topic in optics research. 1450 01:19:49,390 --> 01:19:54,290 Now we know how to design, theoretically, structures. 1451 01:19:54,290 --> 01:19:57,930 By the way, natural materials, they don't have this property. 1452 01:19:57,930 --> 01:20:00,890 So you have to try really hard to make this property happen, 1453 01:20:00,890 --> 01:20:03,650 for example, by patterning a dielectric. 1454 01:20:03,650 --> 01:20:06,570 If you create patterns in a certain way, 1455 01:20:06,570 --> 01:20:08,320 then you can cause-- 1456 01:20:08,320 --> 01:20:11,900 you can make group, I mean dispersion diagrams 1457 01:20:11,900 --> 01:20:13,490 that look like this. 1458 01:20:13,490 --> 01:20:16,310 And, of course, to fabricate them-- 1459 01:20:16,310 --> 01:20:18,017 you know, it is one thing to design them 1460 01:20:18,017 --> 01:20:20,100 and simulate them and one thing to fabricate them. 1461 01:20:20,100 --> 01:20:24,070 So these are all very hot topics in optics research. 1462 01:20:24,070 --> 01:20:26,130 So that's why I-- 1463 01:20:26,130 --> 01:20:29,610 and also, beat is sort of a very important topic. 1464 01:20:29,610 --> 01:20:31,730 So that's why I spend some time talking about. 1465 01:20:36,920 --> 01:20:39,380 And one of the problems-- 1466 01:20:39,380 --> 01:20:42,560 again, in the new homework that I posted, one of the problems 1467 01:20:42,560 --> 01:20:46,340 I'll get you to play a little bit with this dispersion 1468 01:20:46,340 --> 01:20:49,580 relation so you can get a feel for it. 1469 01:20:49,580 --> 01:20:52,910 Unfortunately, in this class, we do mostly optics 1470 01:20:52,910 --> 01:20:54,890 in the space domain. 1471 01:20:54,890 --> 01:20:58,720 This, what I'm discussing here is kind of time domain, right? 1472 01:20:58,720 --> 01:21:02,750 Because we're talking about modulations, and carriers, 1473 01:21:02,750 --> 01:21:03,753 and so on. 1474 01:21:03,753 --> 01:21:05,420 This class is mostly about space domain. 1475 01:21:05,420 --> 01:21:07,340 So we don't spend too much time on this topic. 1476 01:21:07,340 --> 01:21:10,460 But anyway, it is sort of important to know. 1477 01:21:10,460 --> 01:21:12,110 That's why I covered it. 1478 01:21:16,690 --> 01:21:21,300 With that, unless there is any questions, 1479 01:21:21,300 --> 01:21:25,300 I will move on to electromagnetics. 1480 01:21:25,300 --> 01:21:26,291 Questions? 1481 01:21:34,908 --> 01:21:36,200 AUDIENCE: George, one question. 1482 01:21:36,200 --> 01:21:39,050 So just now, we show that carrier and the modulation wave 1483 01:21:39,050 --> 01:21:40,980 actually have different velocity. 1484 01:21:40,980 --> 01:21:43,280 But in everyday life, common sense 1485 01:21:43,280 --> 01:21:45,560 for [INAUDIBLE],, for example, the red light 1486 01:21:45,560 --> 01:21:48,800 and the yellow light travel at the same speed. 1487 01:21:48,800 --> 01:21:51,170 I mean that how different frequency 1488 01:21:51,170 --> 01:21:52,350 have different wavelengths. 1489 01:21:52,350 --> 01:21:56,430 But the product of these two is constant. 1490 01:21:56,430 --> 01:21:59,350 GEORGE BARBASTATHIS: Not at all, for example, we 1491 01:21:59,350 --> 01:22:01,720 know that is not true in glass. 1492 01:22:01,720 --> 01:22:09,134 That is the reason why a prism analyzes white light. 1493 01:22:09,134 --> 01:22:11,990 It is because red and yellow, they 1494 01:22:11,990 --> 01:22:14,840 travel with different velocities in glass. 1495 01:22:14,840 --> 01:22:18,030 So any medium that is dispersive, 1496 01:22:18,030 --> 01:22:19,550 they have different velocities. 1497 01:22:19,550 --> 01:22:22,420 AUDIENCE: So only in a vacuum they have the same velocity. 1498 01:22:22,420 --> 01:22:23,930 GEORGE BARBASTATHIS: In vacuum, yes. 1499 01:22:23,930 --> 01:22:27,200 So the linear dispersion relation 1500 01:22:27,200 --> 01:22:31,315 that I showed before, that is for vacuum. 1501 01:22:41,710 --> 01:22:44,485 Actually, it is also linear in a medium, 1502 01:22:44,485 --> 01:22:45,610 but with a different slope. 1503 01:22:45,610 --> 01:22:47,325 I should have said that. 1504 01:22:47,325 --> 01:22:49,450 COLIN SHEPPARD: I was just going to point out that, 1505 01:22:49,450 --> 01:22:51,340 actually, you show in the phase velocity 1506 01:22:51,340 --> 01:22:54,150 there for the free space, not for the-- 1507 01:22:54,150 --> 01:22:55,900 GEORGE BARBASTATHIS: For free space, yeah. 1508 01:22:55,900 --> 01:22:57,490 COLIN SHEPPARD: Yeah, but the point 1509 01:22:57,490 --> 01:23:02,190 where you've measured the group velocity, that you've 1510 01:23:02,190 --> 01:23:07,270 labeled omega c kc, if you drew a line from the origin 1511 01:23:07,270 --> 01:23:09,895 to that point, the slope of that would give the phase velocity. 1512 01:23:09,895 --> 01:23:11,853 GEORGE BARBASTATHIS: You are correct, actually. 1513 01:23:11,853 --> 01:23:13,680 Yes, yes, yes, thank you, yeah. 1514 01:23:13,680 --> 01:23:16,210 Yes, that's very true. 1515 01:23:16,210 --> 01:23:18,640 The phase velocity is omega over k. 1516 01:23:18,640 --> 01:23:20,200 So it should be that one. 1517 01:23:20,200 --> 01:23:21,800 Yes, so I need to correct the slides. 1518 01:23:21,800 --> 01:23:23,810 Thank you. 1519 01:23:23,810 --> 01:23:26,670 Yeah, this is the phase velocity of free space. 1520 01:23:26,670 --> 01:23:29,430 COLIN SHEPPARD: And it's greater than the speed of light. 1521 01:23:29,430 --> 01:23:30,847 GEORGE BARBASTATHIS: That's right. 1522 01:23:35,433 --> 01:23:36,850 How could that happen, by the way? 1523 01:23:39,773 --> 01:23:41,440 Because this is a phase velocity, right? 1524 01:23:41,440 --> 01:23:42,607 It can do anything it wants. 1525 01:23:42,607 --> 01:23:46,600 It can go faster than the speed of light. 1526 01:23:46,600 --> 01:23:48,070 Because it carries no information. 1527 01:23:48,070 --> 01:23:50,900 It is simply an oscillation. 1528 01:23:50,900 --> 01:23:53,660 The group velocity, as you will see actually-- 1529 01:23:53,660 --> 01:23:56,260 in the homework, I get you to derive the group velocity 1530 01:23:56,260 --> 01:23:57,350 for this case. 1531 01:23:57,350 --> 01:23:59,860 You will see from the answer that the group velocity 1532 01:23:59,860 --> 01:24:02,770 is actually smaller than the speed of light. 1533 01:24:02,770 --> 01:24:06,410 You will see that the group velocity for this diagram, 1534 01:24:06,410 --> 01:24:13,300 it is given by something like c square root 1 minus something. 1535 01:24:13,300 --> 01:24:18,770 So the group velocity is always less than c. 1536 01:24:24,960 --> 01:24:26,970 OK, so I will post the corrected version. 1537 01:24:44,920 --> 01:24:46,241 Any other questions? 1538 01:24:52,500 --> 01:24:55,020 AUDIENCE: So would a patterned metamaterial basically 1539 01:24:55,020 --> 01:24:59,310 be the same thing as a special case of a gradient index 1540 01:24:59,310 --> 01:25:00,750 lens, a GRIN? 1541 01:25:10,127 --> 01:25:11,710 GEORGE BARBASTATHIS: I don't know what 1542 01:25:11,710 --> 01:25:13,150 you mean by a special case. 1543 01:25:13,150 --> 01:25:16,517 I mean, they have quite different properties. 1544 01:25:21,290 --> 01:25:23,690 AUDIENCE: Well, I mean you can maybe 1545 01:25:23,690 --> 01:25:28,523 imagine the idea of the gradient in the dielectric material 1546 01:25:28,523 --> 01:25:30,440 where the gradient and the index of refraction 1547 01:25:30,440 --> 01:25:33,200 sort of creating a negative group velocity 1548 01:25:33,200 --> 01:25:36,530 or some case where you're actually diverting light away 1549 01:25:36,530 --> 01:25:39,290 from what you're trying to image. 1550 01:25:43,130 --> 01:25:46,150 GEORGE BARBASTATHIS: I don't think with a simple gradient 1551 01:25:46,150 --> 01:25:48,360 index you can-- 1552 01:25:48,360 --> 01:25:51,040 OK, it depends on what you mean by gradient. 1553 01:25:51,040 --> 01:25:54,820 If it is a slowly varying gradient like the GRIN optics 1554 01:25:54,820 --> 01:25:59,410 usually, I don't think you can create sufficient dispersion 1555 01:25:59,410 --> 01:26:02,890 to turn the light around. 1556 01:26:02,890 --> 01:26:06,640 If you have subwavelength patterns, 1557 01:26:06,640 --> 01:26:09,610 then yes, because the resulting evanescent waves 1558 01:26:09,610 --> 01:26:11,560 that can adapt in ways that-- 1559 01:26:11,560 --> 01:26:15,115 yeah, I mean, this is the whole area-- yeah, so metamaterials. 1560 01:26:15,115 --> 01:26:16,490 But those, usually the gradients. 1561 01:26:16,490 --> 01:26:17,170 Are huge, right? 1562 01:26:17,170 --> 01:26:21,880 Because within a space of less than the free space wavelength, 1563 01:26:21,880 --> 01:26:24,430 you might have variation of index from 3 to 1, 1564 01:26:24,430 --> 01:26:26,380 and then back to 3, and so on. 1565 01:26:26,380 --> 01:26:28,099 So those are huge gradients. 1566 01:26:36,418 --> 01:26:37,960 AUDIENCE: You say that group velocity 1567 01:26:37,960 --> 01:26:41,802 is the velocity with which the information is traveling? 1568 01:26:41,802 --> 01:26:44,010 GEORGE BARBASTATHIS: Well, that's one interpretation, 1569 01:26:44,010 --> 01:26:44,902 yeah. 1570 01:26:44,902 --> 01:26:46,110 This has been challenged too. 1571 01:26:46,110 --> 01:26:49,200 Because people saw that group velocity in some cases 1572 01:26:49,200 --> 01:26:51,360 can also exceed the speed of light. 1573 01:26:51,360 --> 01:26:57,930 So therefore, even that is questionable. 1574 01:26:57,930 --> 01:27:01,560 But anyway, in most cases, group velocity 1575 01:27:01,560 --> 01:27:05,970 is the velocity at which the modulation travels. 1576 01:27:05,970 --> 01:27:09,870 So the modulation usually denotes information. 1577 01:27:09,870 --> 01:27:11,965 If you exceed the group-- 1578 01:27:11,965 --> 01:27:13,980 if the group velocity, if you manage to make it 1579 01:27:13,980 --> 01:27:17,480 exceed the speed of light, which has been proposed, 1580 01:27:17,480 --> 01:27:19,230 then it means that the modulation actually 1581 01:27:19,230 --> 01:27:20,390 does not carry information. 1582 01:27:20,390 --> 01:27:25,140 It is something that you artificially put in there. 1583 01:27:25,140 --> 01:27:28,770 AUDIENCE: So why we means-- 1584 01:27:28,770 --> 01:27:32,100 what is the advantage of negative group velocity means 1585 01:27:32,100 --> 01:27:35,555 opposite direction information travel? 1586 01:27:35,555 --> 01:27:36,930 GEORGE BARBASTATHIS: Well, it can 1587 01:27:36,930 --> 01:27:39,268 make you a pretty good career and get 1588 01:27:39,268 --> 01:27:40,560 you papers published in Nature. 1589 01:27:45,540 --> 01:27:47,540 No, no, I mean, there's-- 1590 01:27:47,540 --> 01:27:48,050 I'm joking. 1591 01:27:48,050 --> 01:27:50,780 But [INAUDIBLE] it is interesting. 1592 01:27:50,780 --> 01:27:53,180 It is interesting physics and very counter-intuitive. 1593 01:27:53,180 --> 01:27:55,490 That is cool, right? 1594 01:27:55,490 --> 01:27:57,970 But also, the people who work on it, 1595 01:27:57,970 --> 01:28:01,970 they have shown mathematically that you 1596 01:28:01,970 --> 01:28:05,390 can get focusing of light that is sometimes tighter 1597 01:28:05,390 --> 01:28:09,260 than traditional refractive-- 1598 01:28:09,260 --> 01:28:13,730 you can get interesting kinds of dispersion for pulse shaping, 1599 01:28:13,730 --> 01:28:14,540 very narrow-- 1600 01:28:14,540 --> 01:28:17,000 you know, you can do some clever things with it. 1601 01:28:20,270 --> 01:28:23,570 You can slow down light, which is-- 1602 01:28:23,570 --> 01:28:25,500 slow down light means that-- 1603 01:28:25,500 --> 01:28:39,360 if you go back to this diagram, you 1604 01:28:39,360 --> 01:28:44,360 see that as you go towards lower frequencies, 1605 01:28:44,360 --> 01:28:48,590 the group velocity becomes progressively smaller. 1606 01:28:48,590 --> 01:28:51,830 In fact, over here, well, if there's a point where 1607 01:28:51,830 --> 01:28:53,900 it is actually 0-- 1608 01:28:53,900 --> 01:28:55,200 so what does it mean? 1609 01:28:55,200 --> 01:28:56,660 Well, it means that if you really 1610 01:28:56,660 --> 01:29:01,940 had the light pulse, not the fringe of the light, 1611 01:29:01,940 --> 01:29:06,770 not the carrier wave, but the envelope, the modulation, 1612 01:29:06,770 --> 01:29:08,900 if you were to run into this frequency 1613 01:29:08,900 --> 01:29:12,650 or very close to this frequency, that could stop, right? 1614 01:29:12,650 --> 01:29:15,500 It would actually-- or it would move very, very slowly. 1615 01:29:15,500 --> 01:29:17,795 That's the principle of slow light, which actually also 1616 01:29:17,795 --> 01:29:18,920 has potential applications. 1617 01:29:18,920 --> 01:29:20,630 For example, you can store data. 1618 01:29:20,630 --> 01:29:23,625 If you can slow light down, you can create an optical buffer. 1619 01:29:23,625 --> 01:29:26,000 You can wait for something else to happen and then launch 1620 01:29:26,000 --> 01:29:27,740 the wave again. 1621 01:29:27,740 --> 01:29:32,242 So there is things that you can do with this in addition 1622 01:29:32,242 --> 01:29:33,200 to interesting physics. 1623 01:29:47,732 --> 01:29:49,190 It's a fascinating topic, this one. 1624 01:29:49,190 --> 01:29:51,540 So I'm glad you're asking questions. 1625 01:30:03,320 --> 01:30:06,590 AUDIENCE: Yeah, and if you go for a higher frequency, 1626 01:30:06,590 --> 01:30:10,520 we can see that we can excite many modes in the waveguide. 1627 01:30:10,520 --> 01:30:12,370 So all the waveguides will-- 1628 01:30:12,370 --> 01:30:16,123 it means all the modes will have different velocity? 1629 01:30:16,123 --> 01:30:17,540 GEORGE BARBASTATHIS: That's right. 1630 01:30:17,540 --> 01:30:18,110 AUDIENCE: So-- 1631 01:30:18,110 --> 01:30:18,840 GEORGE BARBASTATHIS: Which is bad, right? 1632 01:30:18,840 --> 01:30:21,170 Because it means that if your information 1633 01:30:21,170 --> 01:30:25,400 sample split between those two modes, then they will separate. 1634 01:30:25,400 --> 01:30:27,520 So they will arrive at different times. 1635 01:30:27,520 --> 01:30:28,910 That is called modal dispersion. 1636 01:30:28,910 --> 01:30:32,652 And it is a very serious problem in telecommunications. 1637 01:30:32,652 --> 01:30:34,610 It means that you get distortion in the signal. 1638 01:30:38,550 --> 01:30:40,490 Was that your question, or, I'm sorry, 1639 01:30:40,490 --> 01:30:43,344 I kind of answered your question before you even asked it. 1640 01:31:16,010 --> 01:31:16,940 OK, here they are. 1641 01:31:16,940 --> 01:31:21,880 I'm sure all of you, at some point or another, 1642 01:31:21,880 --> 01:31:25,070 at some moment, probably a moment of a nightmare, 1643 01:31:25,070 --> 01:31:26,580 you saw them, right? 1644 01:31:26,580 --> 01:31:29,510 These are Maxwell's equations. 1645 01:31:29,510 --> 01:31:33,100 So you can read them in the notes. 1646 01:31:33,100 --> 01:31:38,372 I have the two slides preceding these where I sort drew them 1647 01:31:38,372 --> 01:31:39,330 in the traditional way. 1648 01:31:39,330 --> 01:31:42,770 And, of course, the book also has an explanation. 1649 01:31:42,770 --> 01:31:46,420 But I would like to ask you if you remember from your physics 1650 01:31:46,420 --> 01:31:49,060 802, those of you who are at MIT, 1651 01:31:49,060 --> 01:31:53,870 or whatever it was, your basic electromagnetics class, do you 1652 01:31:53,870 --> 01:31:57,080 remember each one of those, what is it telling us, 1653 01:31:57,080 --> 01:31:59,598 and why does it look the way it does? 1654 01:31:59,598 --> 01:32:00,890 Let's start with the first one. 1655 01:32:00,890 --> 01:32:01,807 Does anybody remember? 1656 01:32:01,807 --> 01:32:02,990 What does the first one do? 1657 01:32:02,990 --> 01:32:05,240 So actually, these are-- 1658 01:32:05,240 --> 01:32:08,240 you know, these are the same equations. 1659 01:32:08,240 --> 01:32:11,990 One on the left hand side is written in integral form. 1660 01:32:11,990 --> 01:32:15,440 The right hand side is written in differential form. 1661 01:32:15,440 --> 01:32:18,080 Does anybody want to volunteer and say, 1662 01:32:18,080 --> 01:32:20,060 for example, the first row? 1663 01:32:20,060 --> 01:32:21,750 It is the same equation. 1664 01:32:21,750 --> 01:32:22,990 So what does it tell us? 1665 01:32:22,990 --> 01:32:25,020 AUDIENCE: The-- you want to say? 1666 01:32:27,780 --> 01:32:30,160 GEORGE BARBASTATHIS: Toss a coin. 1667 01:32:30,160 --> 01:32:32,260 AUDIENCE: The first one says, the integral form 1668 01:32:32,260 --> 01:32:40,030 says that the total electric flux passing through a closed 1669 01:32:40,030 --> 01:32:44,920 surface is equal to the total charge enclosed 1670 01:32:44,920 --> 01:32:46,900 inside that surface. 1671 01:32:46,900 --> 01:32:50,200 And the equivalent differential form 1672 01:32:50,200 --> 01:32:55,300 says that the total amount of electric field emerging out 1673 01:32:55,300 --> 01:33:01,030 of a point is equal to the amount of charge at that point. 1674 01:33:01,030 --> 01:33:02,800 GEORGE BARBASTATHIS: That's correct. 1675 01:33:02,800 --> 01:33:07,780 So basically, it is telling you that if you have a charge, then 1676 01:33:07,780 --> 01:33:12,580 you have field lines radiating out of that charge. 1677 01:33:12,580 --> 01:33:15,580 And if you enclose this charge with a surface 1678 01:33:15,580 --> 01:33:21,325 and you compute the integral of the-- 1679 01:33:26,145 --> 01:33:27,520 I don't know why this is a cross. 1680 01:33:27,520 --> 01:33:29,020 By the way, there's another mistake. 1681 01:33:29,020 --> 01:33:30,770 This should have been a dot, not a cross. 1682 01:33:30,770 --> 01:33:37,490 But anyway, if you compute this integral, 1683 01:33:37,490 --> 01:33:42,140 it is determined by the total amount of charge. 1684 01:33:42,140 --> 01:33:44,170 OK, we can go back to the notes now. 1685 01:33:44,170 --> 01:33:45,140 [INAUDIBLE] 1686 01:33:54,840 --> 01:33:57,570 So this is what your colleague just said. 1687 01:33:57,570 --> 01:33:59,000 You have a charge. 1688 01:33:59,000 --> 01:34:02,760 Then you have field lines radiating out. 1689 01:34:02,760 --> 01:34:04,580 And then if you take all of these lines, 1690 01:34:04,580 --> 01:34:10,210 dot product them with the elemental surface, 1691 01:34:10,210 --> 01:34:13,380 and then integrate, then you get the amount of charge. 1692 01:34:13,380 --> 01:34:17,020 And why? 1693 01:34:17,020 --> 01:34:19,770 Why should it be so? 1694 01:34:19,770 --> 01:34:23,930 AUDIENCE: Principle of charge conservation. 1695 01:34:23,930 --> 01:34:27,010 GEORGE BARBASTATHIS: OK, but why do the field lines emanate out 1696 01:34:27,010 --> 01:34:27,690 of the charges? 1697 01:34:33,560 --> 01:34:35,770 AUDIENCE: It goes from positive to-- 1698 01:34:35,770 --> 01:34:37,750 I mean, it originates from positive. 1699 01:34:37,750 --> 01:34:43,150 And we can say that field line represents 1700 01:34:43,150 --> 01:34:46,748 the direction of the force on a-- you need positive charge. 1701 01:34:46,748 --> 01:34:49,040 GEORGE BARBASTATHIS: OK, that's what I was looking for. 1702 01:34:49,040 --> 01:34:51,790 So the field lines are basically-- they tell you, 1703 01:34:51,790 --> 01:34:57,130 exactly like you said, what is the direction that the small 1704 01:34:57,130 --> 01:35:00,040 positive charge-- 1705 01:35:00,040 --> 01:35:00,700 let me repeat. 1706 01:35:00,700 --> 01:35:03,700 What is the direction of the force 1707 01:35:03,700 --> 01:35:05,560 that the small positive charge would 1708 01:35:05,560 --> 01:35:07,750 feel if you were to place it at any given 1709 01:35:07,750 --> 01:35:09,730 position in that space? 1710 01:35:09,730 --> 01:35:13,150 This is what the electric field says. 1711 01:35:13,150 --> 01:35:15,610 And then there is a relationship called 1712 01:35:15,610 --> 01:35:21,090 Coulomb's law, which for two given charges, 1713 01:35:21,090 --> 01:35:22,510 it tells you the force. 1714 01:35:22,510 --> 01:35:24,410 So if you have a single charge here. 1715 01:35:24,410 --> 01:35:26,830 And then you apply Coulomb's law, 1716 01:35:26,830 --> 01:35:29,410 then you get these radiating lines. 1717 01:35:29,410 --> 01:35:31,450 So then Gauss' law is basically nothing other 1718 01:35:31,450 --> 01:35:32,620 than Coulomb's law. 1719 01:35:32,620 --> 01:35:35,170 And yes, it also has to do with charge conservation. 1720 01:35:35,170 --> 01:35:38,500 Because if you somehow cancel all the charges 1721 01:35:38,500 --> 01:35:42,130 inside this volume, then the integral will vanish. 1722 01:35:42,130 --> 01:35:45,400 Because it means that you don't have any net electric-- 1723 01:35:45,400 --> 01:35:47,740 actually, you can have electric charge radiating. 1724 01:35:47,740 --> 01:35:49,020 But the integral will vanish. 1725 01:35:49,020 --> 01:35:55,180 In other words, if you have the charges surrounding, somehow, 1726 01:35:55,180 --> 01:36:01,460 this space, the space cannot pull them all inside. 1727 01:36:01,460 --> 01:36:04,110 OK, what about-- you have a question? 1728 01:36:07,270 --> 01:36:08,380 What about the second one? 1729 01:36:22,320 --> 01:36:24,050 AUDIENCE: That means, [INAUDIBLE] 1730 01:36:24,050 --> 01:36:27,903 line integral is 0 means b is a closed line. 1731 01:36:27,903 --> 01:36:29,570 GEORGE BARBASTATHIS: B is a closed line. 1732 01:36:29,570 --> 01:36:33,740 And b is a closed line why? 1733 01:36:33,740 --> 01:36:37,310 AUDIENCE: It means that there is no magnetic charge. 1734 01:36:37,310 --> 01:36:39,030 GEORGE BARBASTATHIS: That's right. 1735 01:36:39,030 --> 01:36:41,390 Magnets only come as dipoles. 1736 01:36:41,390 --> 01:36:45,670 You cannot have an isolated north pole, for example, 1737 01:36:45,670 --> 01:36:46,760 of a magnet. 1738 01:36:46,760 --> 01:36:50,105 So actually, this is the same equation. 1739 01:36:50,105 --> 01:36:51,980 In fact, they're also known by the same name. 1740 01:36:51,980 --> 01:36:55,080 They're known as Gauss' law. 1741 01:36:55,080 --> 01:36:59,240 This type of equation, except there's 1742 01:36:59,240 --> 01:37:00,650 no such thing as magnetic charge. 1743 01:37:03,190 --> 01:37:05,440 What about the next one? 1744 01:37:05,440 --> 01:37:08,575 This one looks awful, del cross c equals db dt. 1745 01:37:17,330 --> 01:37:18,580 So let's look at the integral. 1746 01:37:18,580 --> 01:37:23,550 Usually the differential form is mathematically better but not 1747 01:37:23,550 --> 01:37:26,970 very insightful physically. 1748 01:37:26,970 --> 01:37:31,380 So one gets more information by looking at the integral forms, 1749 01:37:31,380 --> 01:37:34,440 even though the integral forms look really frightening. 1750 01:37:34,440 --> 01:37:41,640 But anyway, the third equation is known as Faraday's law. 1751 01:37:41,640 --> 01:37:49,500 And it says that if you have a viable magnetic field 1752 01:37:49,500 --> 01:38:01,330 inside a wire, then a potential will develop across this wire. 1753 01:38:05,570 --> 01:38:08,920 And do you know of any places where this is used? 1754 01:38:14,560 --> 01:38:16,252 Button. 1755 01:38:16,252 --> 01:38:17,210 Even I cannot hear you. 1756 01:38:17,210 --> 01:38:18,460 Imagine the people in Boston. 1757 01:38:22,550 --> 01:38:23,050 Not working? 1758 01:38:26,750 --> 01:38:28,360 Oh, maybe use someone else's. 1759 01:38:32,150 --> 01:38:33,670 OK, [INAUDIBLE], you're on. 1760 01:38:33,670 --> 01:38:36,892 AUDIENCE: It's induction motors and-- 1761 01:38:36,892 --> 01:38:39,350 GEORGE BARBASTATHIS: Yeah, voice coils and generators, they 1762 01:38:39,350 --> 01:38:40,560 use the same principle. 1763 01:38:40,560 --> 01:38:43,250 So, for example, if you-- 1764 01:38:43,250 --> 01:38:45,470 people do this in hydroelectric plants. 1765 01:38:45,470 --> 01:38:47,600 You have a giant magnet. 1766 01:38:47,600 --> 01:38:53,370 And you somehow move a coil inside that magnet. 1767 01:38:53,370 --> 01:38:55,370 And, of course, you have to arrange it properly. 1768 01:38:55,370 --> 01:38:57,200 Then the coil will develop a voltage. 1769 01:38:57,200 --> 01:38:58,740 So that's the principle of-- 1770 01:38:58,740 --> 01:39:01,220 of course, you have to move the coil, for example, 1771 01:39:01,220 --> 01:39:05,370 by pouring water over it, or by steam, or whatever. 1772 01:39:05,370 --> 01:39:10,680 But anyway, that's the principle of a generator. 1773 01:39:10,680 --> 01:39:15,430 OK, so and the last equation that-- 1774 01:39:15,430 --> 01:39:18,710 AUDIENCE: George, just as an aside, 1775 01:39:18,710 --> 01:39:25,580 how does a nuclear power plant generate energy, electricity? 1776 01:39:25,580 --> 01:39:26,990 Does it rotate something? 1777 01:39:26,990 --> 01:39:28,436 GEORGE BARBASTATHIS: Actually, I don't know. 1778 01:39:28,436 --> 01:39:30,620 COLIN SHEPPARD: I think it just generates steam, and then you-- 1779 01:39:30,620 --> 01:39:31,010 AUDIENCE: Oh, OK. 1780 01:39:31,010 --> 01:39:32,360 GEORGE BARBASTATHIS: Then you move a coil, I guess, yeah. 1781 01:39:32,360 --> 01:39:33,485 AUDIENCE: Right, right, OK. 1782 01:39:42,080 --> 01:39:44,450 GEORGE BARBASTATHIS: And then the last one is actually 1783 01:39:44,450 --> 01:39:46,730 the most interesting one. 1784 01:39:46,730 --> 01:39:51,110 Because this law actually was involved with a breakthrough 1785 01:39:51,110 --> 01:39:52,940 in electromagnetics. 1786 01:39:52,940 --> 01:40:05,380 It can be known for a long time that if you have a current, 1787 01:40:05,380 --> 01:40:07,880 a magnetic field develops around that current. 1788 01:40:07,880 --> 01:40:10,370 That is actually Lorentz-- 1789 01:40:10,370 --> 01:40:12,020 it is known as Lorentz force. 1790 01:40:12,020 --> 01:40:17,420 If you have a moving charge, and then you put a little magnet 1791 01:40:17,420 --> 01:40:20,082 nearby, then the-- 1792 01:40:20,082 --> 01:40:21,680 what am I saying. 1793 01:40:21,680 --> 01:40:23,180 Anyway, if you have a moving charge, 1794 01:40:23,180 --> 01:40:29,870 and if you have a current, it creates a magnetic field 1795 01:40:29,870 --> 01:40:30,370 around it. 1796 01:40:30,370 --> 01:40:32,930 That has been known for a long time. 1797 01:40:32,930 --> 01:40:35,360 But what was not known and was noticed 1798 01:40:35,360 --> 01:40:41,790 by Maxwell was that if you have a capacitor-- 1799 01:40:41,790 --> 01:40:44,590 well, in a capacitor, you can not have a current per se. 1800 01:40:44,590 --> 01:40:47,580 Because the capacitor, well, inside there's a dielectric. 1801 01:40:47,580 --> 01:40:49,800 So there can be no current. 1802 01:40:49,800 --> 01:40:55,360 But you can have a variable charge 1803 01:40:55,360 --> 01:40:57,760 density in the capacitor plate. 1804 01:40:57,760 --> 01:41:02,650 So what I'm trying to say is that if you apply a-- 1805 01:41:02,650 --> 01:41:05,050 think of charge in a capacitor. 1806 01:41:05,050 --> 01:41:10,660 When you charge a capacitor, the capacitor, at the beginning, 1807 01:41:10,660 --> 01:41:13,840 the voltage across the capacitor is 0. 1808 01:41:13,840 --> 01:41:15,820 And by the time you have totally charged it, 1809 01:41:15,820 --> 01:41:19,590 the voltage has reached some value. 1810 01:41:19,590 --> 01:41:22,560 Well, while the voltage is changing, 1811 01:41:22,560 --> 01:41:27,660 there is actually charge flow into the capacitor. 1812 01:41:27,660 --> 01:41:31,080 There is a positive charge going to the positive electrode, 1813 01:41:31,080 --> 01:41:36,600 then negative leaving from the opposite electrode. 1814 01:41:36,600 --> 01:41:38,280 And at the same time, what is happening, 1815 01:41:38,280 --> 01:41:40,860 the electric field inside the capacitor grows. 1816 01:41:40,860 --> 01:41:43,830 Because as you get more charge accumulating in the capacitor 1817 01:41:43,830 --> 01:41:48,340 plates, then the electric field of the capacitor is changing. 1818 01:41:48,340 --> 01:41:52,110 So in a sense, you can think of it as a current as well. 1819 01:41:52,110 --> 01:41:55,860 And what Maxwell-- he guessed it, actually. 1820 01:41:55,860 --> 01:41:58,770 He didn't observe it. 1821 01:41:58,770 --> 01:42:00,300 But he guessed it. 1822 01:42:00,300 --> 01:42:05,100 He guessed that this kind of variable electric field 1823 01:42:05,100 --> 01:42:09,160 should also generate a magnetic field. 1824 01:42:09,160 --> 01:42:12,120 And the only reason he guessed it is by symmetry 1825 01:42:12,120 --> 01:42:14,370 from the previous law. 1826 01:42:14,370 --> 01:42:15,280 What does that say? 1827 01:42:15,280 --> 01:42:19,020 It says that if you have a variable magnetic flux, 1828 01:42:19,020 --> 01:42:21,990 you generate an electrical potential. 1829 01:42:21,990 --> 01:42:23,700 Well, he looked at this equation. 1830 01:42:23,700 --> 01:42:25,070 And he said, wait a minute. 1831 01:42:25,070 --> 01:42:27,350 It's fine that if I have a moving charge, 1832 01:42:27,350 --> 01:42:28,470 I have a magnetic field. 1833 01:42:28,470 --> 01:42:29,370 OK, fine. 1834 01:42:29,370 --> 01:42:31,080 But if also I have a moving-- 1835 01:42:31,080 --> 01:42:34,020 if I have a variable potential, then why 1836 01:42:34,020 --> 01:42:36,090 should I not generate a magnetic field? 1837 01:42:36,090 --> 01:42:37,890 I should. 1838 01:42:37,890 --> 01:42:40,210 And it turned out to be true. 1839 01:42:40,210 --> 01:42:43,900 I mean-- Maxwell, by the way, he was-- 1840 01:42:43,900 --> 01:42:46,910 do you know what was his day job? 1841 01:42:46,910 --> 01:42:55,610 James Clerk Maxwell, he was a fluid mechanic. 1842 01:42:55,610 --> 01:42:57,640 He was [INAUDIBLE]. 1843 01:42:57,640 --> 01:43:00,670 And that's why the terminology-- not the terminology, 1844 01:43:00,670 --> 01:43:03,270 but the notation in electromagnetic magnetics 1845 01:43:03,270 --> 01:43:07,150 is actually very similar to the fluid mechanics notation, rho, 1846 01:43:07,150 --> 01:43:08,513 and epsilon, and all that stuff. 1847 01:43:08,513 --> 01:43:10,430 I mean, they're not the same things, but the-- 1848 01:43:13,000 --> 01:43:18,500 and also, these theorems, the mathematical theorems, 1849 01:43:18,500 --> 01:43:20,660 they come from fluid mechanics. 1850 01:43:20,660 --> 01:43:22,430 They apply to incomprehensible flows. 1851 01:43:22,430 --> 01:43:24,200 For example, the Gauss theorem, it 1852 01:43:24,200 --> 01:43:34,960 applies to incomprehensible flow in the pressure potential. 1853 01:43:34,960 --> 01:43:38,270 Anyway, so Maxwell did that first. 1854 01:43:38,270 --> 01:43:45,520 So he guessed that you should have this form in here. 1855 01:43:45,520 --> 01:43:47,860 And then he actually wrote down all of these equations 1856 01:43:47,860 --> 01:43:50,970 in a coordinated form. 1857 01:43:50,970 --> 01:43:53,500 And the reason this was a very valuable thing, 1858 01:43:53,500 --> 01:43:58,340 and I guess we're all employed because of this [INAUDIBLE],, 1859 01:43:58,340 --> 01:44:02,803 is because if you manipulate Maxwell's equations-- 1860 01:44:02,803 --> 01:44:04,970 and I will not do it now, because I ran out of time. 1861 01:44:04,970 --> 01:44:07,500 But if you manipulate them, you can actually 1862 01:44:07,500 --> 01:44:11,680 show that you arrive at the wave equation for the electric 1863 01:44:11,680 --> 01:44:12,880 or the magnetic field. 1864 01:44:12,880 --> 01:44:14,710 I did it here for the electric field. 1865 01:44:14,710 --> 01:44:17,590 But you can also derive the same equation 1866 01:44:17,590 --> 01:44:19,267 for the magnetic field. 1867 01:44:19,267 --> 01:44:21,100 So therefore, what this means is that if you 1868 01:44:21,100 --> 01:44:28,850 have electric and magnetic fields which change in time, 1869 01:44:28,850 --> 01:44:30,590 Maxwell's equations tell you that they 1870 01:44:30,590 --> 01:44:32,960 change in a coordinated way. 1871 01:44:32,960 --> 01:44:36,890 And the way they change is as an electromagnetic wave. 1872 01:44:47,110 --> 01:44:48,340 I think it's time to quit. 1873 01:44:48,340 --> 01:44:51,300 In fact, we're already five minutes late, so.